diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfcky" "b/data_all_eng_slimpj/shuffled/split2/finalzzfcky" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfcky" @@ -0,0 +1,5 @@ +{"text":"\\section{introduction}\\label{sec1}\n\n\nIn standard cosmology, the Copernican principle is applied i.e., it is assumed\nthat we are not living at a privileged position in the universe. \nCombining the Copernican principle with the isotropy of \nthe Cosmic Microwave Background (CMB) radiation, \nleads to the conclusion that our universe is well described by the\nhomogeneous and isotropic universe model. \nIn the framework of the homogeneous and isotropic universe, \nthe observational data of the luminosity distances of Type Ia supernovae (SNIa) \nindicates an acceleration of the cosmic volume expansion of our universe. \nThe acceleration of the cosmic volume expansion in the homogeneous and \nisotropic universe implies the existence of \nso-called dark energy that acts as a source of a repulsive gravitational force\nif we assume general relativity at cosmological scales.\nAt present, there is no theory that can naturally explain the origin of dark energy, \nand it seems worth investigating alternative scenarios. \nIn order to do this, we have to discard general relativity or the homogeneity \nassumption. \n\nInhomogeneous cosmological models without dark energy\nhave been proposed independently \nby Tomita~\\cite{Tomita:1999qn,Tomita:2000jj,Tomita:2001gh} \nand C\\'el\\'erier~\\cite{Celerier:1999hp}. \nIn C\\'el\\'erier's model, the observer is located at the symmetry center \nof a very large spherical void which can explain the SNIa observations. \nSince the observer is located at\na special position in the universe, we call this model \na ``the non-Copernican universe model\" in this paper. \nThe common assumption of non-Copernican universe models \nis that an observer is located \nin the vicinity of the symmetry center,\nwhich explains the fact that the CMB radiation is observed to be isotropic. \nThe most common way to describe \nnon-Copernican universe models \nis to use the Lema\\^{\\i}tre-Tolman-Bondi(LTB) solution for the Einstein equations, \nwhich describes the motion of spherically symmetric dust. \n\nSNIa observations in non-Copernican universe models have been\nstudied by many researchers~\\cite{Tomita:1999qn,Tomita:2000jj,Tomita:2001gh,\nCelerier:1999hp,Goodwin:1999ej,Clifton:2008hv,\nIguchi:2001sq,Vanderveld:2006rb,Yoo:2008su,Celerier:2009sv,Kolb:2009hn,Yoo:2010qn}, \nand it has been proven that the distance-redshift relation \nin the $\\Lambda$CDM universe can be reproduced using LTB universe \nmodels~\\cite{Mustapha:1998jb,Yoo:2008su,Celerier:2009sv,Kolb:2009hn}. \nNon-Copernican universe models have been tested by other observations including \nthe CMB acoustic \npeaks~\\cite{Alnes:2005rw,Alexander:2007xx,Zibin:2008vk,\nGarciaBellido:2008nz,Yoo:2010qy,Clarkson:2010ej,\nMarra:2010pg,Moss:2010jx,Biswas:2010xm,Marra:2011ct,Nadathur:2010zm}, \nthe radial baryon acoustic oscillation scales~\\cite{Zibin:2008vk,GarciaBellido:2008yq,Zumalacarregui:2012pq},\nthe kinematic Suniyaev-Zeldovich \neffect~\\cite{Bull:2011wi,GarciaBellido:2008gd,Yoo:2010ad,Zhang:2010fa,Moss:2011ze} \nand others~\\cite{Alnes:2006pf,Alnes:2006uk,Bolejko:2005fp,Dunsby:2010ts,\nEnqvist:2009hn,Enqvist:2006cg,Goto:2011ru,Kodama:2010gr,\nQuartin:2009xr,Regis:2010iq,Romano:2009mr,Romano:2010nc,Romano:2011mx,Tanimoto:2009mz,\nUzan:2008qp,Yoo:2010hi,Zibin:2011ma}.\nAlthough these observations have imposed restrictions on these models, \nthey have not yet ruled out the models. It is not easy to confirm whether our universe \nfollows the Copernican principle. \n\n\nIn this paper, we focus on the evolution of structures such as clusters of galaxies \nand super-clusters in the non-Copernican universe in the matter \ndominant era which is well described by the LTB solution. \nIt is expected that observations of the large-scale structures and \ntheir evolution can be used to test the non-Copernican universe model, \nsince the evolution of the anisotropic perturbations reflects the tidal force\nin the background spacetime. \nHowever, the evolution of perturbations in \nthe LTB solution has not yet been fully studied. \nThis is because the isometries in the LTB spacetime are less than \nin the homogeneous and isotropic universe. \nAlthough master equations for perturbations for \ngeneral spherically symmetric spacetimes \nhave been derived a long time ago~\\cite{Gerlach:1980}\n(see also Ref.~\\cite{Clarkson:2009sc} for the LTB background), \nthese equations for the LTB solution cannot be reduced \nto ordinary differential equations, in general.\nThis is a very different situation from the case of the homogeneous \nand isotropic universe. \n\nRecently, Alonso et al.~\\cite{Alonso:2010zv} performed \nnumerical simulations for non-Copernican models including only cold dark matter. \nThey studied the perturbed Einstein-deSitter universe with \ntwo kinds of perturbations: one forms a spherical void, \nand the other is a non-spherical perturbation with a power spectrum with a random phase \nGaussian probability distribution. \nThey followed the growth of these perturbations using Newtonian $N$-body simulations.\nHowever, in order to confirm the validity of the tnumerical simulations, analytic complementary\nstudies are necessary. \nSome authors~\\cite{Moss:2010jx,Zibin:2008vj,Dunsby:2010ts} have studied perturbations\nby using a ``silent approximation'' that neglects the magnetic part of the Weyl tensor.\nHere, we should note that the magnetic part of the Weyl tensor usually plays an important role \neven in Newtonian situations~\\cite{Bertschinger:1994nc}. \nHence, in this paper, we propose another complementary analytic approach. \n\nIn many non-Copernican models, \nthe void structure becomes nonlinear at the present time. \nHowever, Enqvist et al.~\\cite{Enqvist:2009hn} pointed out that \nthe void inhomogeneity remains in a quasi-linear regime \n$\\sim \\mathcal{O}(0.1)$ inside a past light-cone of an observer at the center of the void. \nActually, they considered a linear perturbation in \nthe Einstein-deSitter universe that is consistent with the SNIa data, \nand showed that the fraction of the spherically symmetric \nlinear perturbation does not exceed 30\\% inside the past light-cone. \nThis result implies that non-Copernican LTB cosmological models compatible with \nthe observed distance-redshift relation \nmay be studied by perturbation theory for the homogeneous and \nisotropic universe filled with dust at least for the inside of \nthe past light-cone of the central observer. \n\nIn this paper, we investigate the growth of perturbations in the \nnon-Copernican universe models by applying the above idea. \nIt is rather difficult to analyze the evolution of anisotropic perturbations \nin the non-Copernican LTB universe model, while it is much easier to study the evolution of \nnon-linear perturbations in the homogeneous and isotropic universe model \nby successive approximation. \nWe adopt the latter approach. We introduce two-parameter perturbations with \nsmall expansion parameters $\\kappa$ and $\\epsilon$\nin a homogeneous and isotropic dust universe. \nThe limit $\\epsilon\\rightarrow0$ leads to the exact \nLTB solution, if we take all orders of $\\kappa$ into account. By contrast, the limit \n$\\kappa\\rightarrow0$ with $0<\\epsilon\\ll1$ leads to the homogeneous and \nisotropic universe with small anisotropic perturbations. \nThen, in order to see the effect of the void structure \non the evolution of the anisotropic perturbations, we study the non-linear \neffects up to the order of $\\kappa\\epsilon$, following Ref.~\\cite{Tomita:1967}. \n\nThis paper is organized as follows. \nIn \\S~\\ref{sec2}, we derive the equations for \nperturbations parametrized by $\\kappa$ and $\\epsilon$ \nin the homogeneous and isotropic dust universe and obtain general solutions \nup to order $\\kappa\\epsilon$. \nIn \\S~\\ref{sec3}, by fixing the initial conditions, \nwe calculate the angular power spectrum of the density perturbations. \nIn \\S~\\ref{sec4}, we analyze the growth of the perturbations \nby using the angular growth rate. \n\\S~\\ref{sec5} is devoted to a summary and discussion.\n\n\nIn this paper, we use the geometrized units in which \nthe speed of light and Newton's gravitational constant are one, respectively. \nThe Latin indices denote the spatial components, whereas the Greek indices represent\nthe spacetime components. \n \n\\section{two-parameter perturbations in a homogeneous and isotropic dust universe}\n\\label{sec2}\n\\subsection{Perturbations with two kinds of parameters}\nAs mentioned, we study the perturbations in the homogeneous and isotropic universe which is \noften called the Friedmann-Lema\\^{\\i}tre-Robertson-Walker (FLRW) universe. \nSince the structure formation begins after the universe has begun to be dominated by non-relativistic matter, \nit is sufficient for our purpose to consider the universe model filled with dust. \nUsing the spherical polar coordinates for 3-dimensional space, the line-element is given by\n\\begin{eqnarray}\n d\\bar{s}^2 &=& -dt^2+a^2(t)\\left(\\frac{dr^2}{1-Kr^2}+r^2d\\Omega^2\\right) =:-dt^2+a^2(t)\\gamma_{ij}dx^idx^j,\n \\end{eqnarray}\nwhere $a(t)$ is the scale factor which will be determined by the Einstein equations, \n$K$ is constant, $d\\Omega^2$ is the 2-dimensional round metric, \nand, for later convenience, we have defined the background conformal 3-metric $\\gamma_{ij}$. \nThe constant $K$ has the same sign as that of the curvature of the 3-dimensional \nspace specified by $t=$constant.\nThe stress-energy tensor of dust is given by\n\\begin{equation}\n \\bar T^{\\mu \\nu}= \\bar{\\rho }(t)\\bar{u}^\\mu \\bar{u}^\\nu ,\n\\end{equation}\nwhere $\\bar{\\rho}(t)$ is the energy density, \nand $\\bar{u}^\\mu$ is the 4-velocity whose components are \ngiven by $\\bar{u}^\\mu=(1,0,0,0)$. \nThe Einstein equations for the FLRW universe are \n\\begin{eqnarray}\n \\left(\\frac{\\dot{a}}{a}\\right)^2=\\frac{8\\pi}{3}\\bar{\\rho }-\\frac{K}{a^2} \\quad{\\rm and}\\quad\n \\frac{\\ddot{a}}{a}=-\\frac{4\\pi}{3}\\bar{\\rho},\n \\label{bac1}\n\\end{eqnarray}\nwhere a dot denotes differentiation with respect to $t$. \n\nSince our main interest is the evolution of density contrasts \nand their correlations, \nwe consider only scalar perturbations in the FLRW universe. \nAs mentioned in the previous section, \nwe introduce two small independent non-negative parameters, $\\kappa $ and $\\epsilon $. \nThe limit $\\epsilon\\rightarrow0$ leads to the exact \nLTB solution, if we take all the orders of $\\kappa$ into account. By contrast, the limit \n$\\kappa\\rightarrow0$ with $0<\\epsilon\\ll1$ leads to the homogeneous and \nisotropic universe with small anisotropic perturbations. \n\nThen, by choosing the synchronous comoving gauge, \nthe line element of the perturbed spacetime can be written in the form\n\\begin{eqnarray}\n ds^2\n&=&\n -dt^2+a^2(t)\\sum_{N=0}\\kappa^N\\biggl[\nl^{(N)}_{\\parallel}(t,r)\\frac{dr^2}{1-Kr^2}+l^{(N)}_{\\bot }(t,r)r^2d\\Omega^2 \\cr\n&+&\\epsilon\\Big(A^{(N+1)}(t,r,{\\bf \\Omega})\\gamma _{ij}\n+\\mathcal{D}_i\\mathcal{D}_jB^{(N+1)}(t,r, {\\bf \\Omega})\\Big)dx^idx^j \\cr\n&+&{\\cal O}(\\epsilon^2)\\biggr],\n\\label{ds2sec}\n\\end{eqnarray}\nwhere $l_{\\parallel}^{(0)}=l_{\\bot}^{(0)}=1$, $\\bf \\Omega=(\\theta,\\phi)$ \nare the polar and azimuthal angles, and $\\mathcal{D}_i$ denotes the covariant derivative \nwith respect to $\\gamma_{ij}$. \nThe perturbed stress-energy tensor is given by\n\\begin{equation}\n T^{\\mu \\nu }\n =\n \\bar{\\rho }(t)\\bar{u}^\\mu \\bar{u}^\\nu \n\\sum_{N=0}\\kappa^N\\left[\\Delta ^{(N)}(t,r)+\\epsilon \\delta ^{(N+1)}(t,r,{\\bf \\Omega})\n+{\\cal O}(\\epsilon^2)\\right],\n \\label{tmnsec}\n\\end{equation}\nwhere $\\Delta^{(0)}=1$. \nIf we wish to study the evolution of the perturbed FLRW universe \nwith the same accuracy as the linear perturbation analysis for the LTB solution, \nwe should take all orders of $\\kappa$ and the first order with respect to $\\epsilon$. \nHowever, if $\\kappa$ is much smaller than unity, \nit will be possible to evaluate the evolution of the anisotropic perturbations in the \nLTB solution by studying up to the first order with respect to $\\kappa$.\nIn this approximation,\nthe effect of the void structure on the evolution of anisotropic perturbations appears \nat order $\\kappa\\epsilon$. \n\n\\subsection{First order perturbations}\nWe expand the perturbation variables in terms of the\nspherical harmonic functions $Y_{\\ell m}({\\bf \\Omega})$.\nThe Einstein equations of order $\\kappa$ correspond to the equations for the \nperturbations of $\\ell=0$ mode: \n\\begin{eqnarray}\n \\ddot{\\Delta}^{(1)}+2H\\dot{\\Delta}^{(1)}-4\\pi \\bar{\\rho}\\Delta^{(1)} &=&0,\n \\label{kap1}\n \\\\\n \\dot{l}^{(1)}_{\\parallel}-(r\\dot{l}^{(1)}_\\bot )^{'}&=&0,\n \\label{kap2}\n \\\\\n \\dot{l}^{(1)}_{\\parallel}+2\\dot{l}^{(1)}_\\bot &=&-2\\dot{\\Delta}^{(1)},\n \\label{kap3}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nH:=\\frac{\\dot{a}}{a}, \n\\end{equation}\nand a dash denotes a partial differentiation with respect to $r$. \nThe Einstein equations of order $\\epsilon$ lead to the equations for the \nperturbations of $\\ell>0$ modes, and we obtain\n\\begin{eqnarray}\n \\dot{A}_{\\ell m}^{(1)}(t,r)-K\\dot{B}_{\\ell m}^{(1)}(t,r)&=&0,\n \\label{eps2}\n \\\\\n (1-Kr^2)\\dot{B}_{\\ell m}^{(1)}{}''(t,r)\n +\\biggl[\\frac{2(1-Kr^2)}{r}-Kr\\biggr]\\dot{B}_{\\ell m}^{(1)}{}'(t,r)&& \\cr\n +\\biggl[3K-\\frac{\\ell (\\ell +1)}{r^2}\\biggr]\\dot{B}_{\\ell m}^{(1)}(t,r) \n &=&-2\\dot{\\delta}_{\\ell m}^{(1)}(t,r),\n \\label{eps3} \\\\\n \\ddot{\\delta}_{\\ell m}^{(1)}(t,r)+2H\\dot{\\delta}_{\\ell m}^{(1)}(t,r)-4\\pi \\bar{\\rho }\\delta_{\\ell m}^{(1)}(t,r)&=&0,\n \\label{eps1}\n\\end{eqnarray}\nwhere we have used the eigenvalue equation\n\\begin{eqnarray}\n & & \n \\left(\n \\frac{\\partial^2}{\\partial\\theta ^2}+\\frac{\\cos \\theta }{\\sin \\theta }\\frac{\\partial}{\\partial\\theta}\n +\\frac{1}{\\sin^2\\theta }\\frac{\\partial^2}{\\partial\\phi ^2}\n \\right)\n Y_{\\ell m}({\\bf \\Omega})=-\\ell (\\ell +1)Y_{\\ell m}({\\bf \\Omega}).\n \\label{eigen}\n\\end{eqnarray}\nWe note that the equations of order $\\epsilon$ and $\\kappa$ \ndecouple with each other. \nFor later convenience, we also show the equation for $B(t,r,{\\bf \\Omega})$ before deriving \nEq.~\\eqref{eps3} by the spherical harmonics expansion:\n\\begin{eqnarray}\n & &\n \\big(\\mathcal{D}^i\\mathcal{D}_i+3K\\big)\\dot{B}^{(1)}(t,r,{\\bf \\Omega})\n =-2\\dot{\\delta}^{(1)}(t,r,{\\bf \\Omega}). \n \\label{eps4}\n\\end{eqnarray}\n\nGeneral solutions for Eqs.~\\eqref{kap1} and \\eqref{eps1} are given by\n\\begin{eqnarray}\n \\Delta ^{(1)}(t,r)\n &=&\n D^+(t)\\Delta_+^{(\\rm i)}(r)+D^-(t)\\Delta_-^{(\\rm i)}(r),\n \\label{Del1}\n \\\\\n \\delta_{\\ell m}^{(1)}(t,r)\n &=&\n D^+(t)\\delta^{(\\rm i)+}_{\\ell m}(r)+D^-(t)\\delta^{(\\rm i)-}_{\\ell m}(r),\n \\label{del1}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n & & \n D^+(t)=H\\int^{a(t)}\\frac{da}{a^3H^3}\\quad{\\rm and}\\quad D^-(t)=H, \n \\label{grow1}\n\\end{eqnarray}\nand \n$\\Delta^{(\\rm i)}_{\\pm }$ and $\\delta^{(\\rm i)\\pm }_{\\ell m}$ stand for initial values.\n$D^+$ and $D^-$ represent the growing and decaying modes, respectively.\n\n\\subsection{The perturbations of order $\\kappa\\epsilon$}\nAs already mentioned, we are interested in \nthe effect of the void structure on the evolution of \nanisotropic linear perturbation, and this effect first appears at \norder $\\kappa\\epsilon$. Hence we shall focus on the perturbations of this order. \n\nThe perturbations of order $\\kappa\\epsilon$ correspond to the second order \nperturbations in the FLRW universe model. From the second-order Einstein equations together with \nthe background and the linearized Einstein equations,\nwe obtain the evolution equations for the expansion coefficients of $\\delta^{(2)}(t,r,{\\bf \\Omega})$ \nwith respect to $Y_{\\ell m}({\\bf \\Omega})$ as follows:\n\\begin{eqnarray}\n & &\n \\ddot{\\delta}^{(2)}_{\\ell m}(t,r)+2H\\dot{\\delta}^{(2)}_{\\ell m}(t,r)-4\\pi \\bar{\\rho }(t)\\delta ^{(2)}_{\\ell m}(t,r)\n =\n S_{\\ell m}(t,r),\n \\label{second1}\n\\end{eqnarray}\nwhere the source term $S_{\\ell m}(t,r)$ is given by\n\\begin{eqnarray}\n S_{\\ell m}(t,r)\n &=&\n \\left(\\dot{l}^{(1)}_\\bot -\\dot{l}^{(1)}_{||}\\right)\\left(K-\\frac{\\ell (\\ell +1)}{2r^2}\\right)\\dot{B}_{\\ell m}^{(1)}\n +\\left(\\dot{l}^{(1)}_\\bot -\\dot{l}^{(1)}_{||}\\right)\\frac{(1-Kr^2)}{r}\\dot{B}_{\\ell m}^{(1)}{}'\n \\nonumber \\\\\n & &\n +\\left(2\\dot{\\Delta}^{(1)}-\\dot{l}^{(1)}_{||}\\right)\\dot{\\delta}^{(1)}_{\\ell m}\n +8\\pi \\bar{\\rho}\\Delta ^{(1)}\\delta ^{(1)}_{\\ell m}. \n\\end{eqnarray}\nBy solving Eq.~\\eqref{second1}, we obtain \n\\begin{eqnarray}\n \\delta _{\\ell m}^{(2)}(t,r)\n =\n\\int ^t_{t_{\\rm i}}S_{\\ell m}(s,r)\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)ds\n\\end{eqnarray}\nwhere $W(s)$ is the Wronskian given by $W(s)=D^+(s)\\dot{D}^-(s)-\\dot{D}^+(s)D^-(s)$, \nand homogeneous solutions have been absorbed in $\\delta_{\\ell m}^{(1)}$. \nThen, we obtain the anisotropic linear density contrast $\\delta_{\\ell m}$ \nin the LTB solution as\n\\begin{eqnarray}\n \\delta_{\\ell m}(t,r)\n &=&\n \\epsilon \\delta^{(1)}_{\\ell m}(t,r)\n +\\kappa \\epsilon\n \\int ^t_{t_{\\rm i}}S_{\\ell m}(s,r)\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)ds \\cr\n &+&{\\cal O}(\\kappa^2\\epsilon),\n \\label{dens2}\n\\end{eqnarray}\nwhere $t_{\\rm i}$ is the initial time.\n\n\nHereafter, we neglect the decaying modes of order $\\epsilon$. \nBy using Eqs.~\\eqref{eps3} and \\eqref{del1}, $\\dot{B}_{\\ell m}^{(1)}$ is \nwritten in the form \n\\begin{eqnarray}\n \\dot{B}_{\\ell m}^{(1)}(t,r)=\\dot{D}^+(t)B^{(\\rm i)+}_{\\ell m}(r),\n \\label{B1}\n\\end{eqnarray}\nwhere $B^{(\\rm i)+}_{\\ell m}(r)$ is the initial value.\nThen, the anisotropic density contrast \\eqref{dens2} is rewritten as\n\\begin{eqnarray}\n \\delta _{\\ell m}(t,r)\n &=&\n \\epsilon D^+(t)\\delta^{(\\rm i)+}_{\\ell m}(r) \\cr\n &+&\\kappa \\epsilon \\left[T_1(t,r)\\delta^{(\\rm i)+}_{\\ell m}(r)\n +T_2(t,r,\\ell )B^{(\\rm i)+}_{\\ell m}(r)\n + T_3(t,r)B^{(\\rm i)+}_{\\ell m}{}'(r)\n \\right] \\cr\n &+&{\\cal O}(\\kappa^2\\epsilon),\n \\label{dens3}\n\\end{eqnarray}\nwhere $T_1$, $T_2$ and $T_3$ are defined by\n\\begin{eqnarray}\n T_1(t,r)\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\left[\\dot{D}^+(s)\\left(2\\dot{\\Delta}^{(1)}(s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\n +D^+(s)\\times 8\\pi \\bar{\\rho}(s)\\Delta ^{(1)}(s,r)\n \\right],\n \\\\\n T_2(t,r,\\ell )\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\n \\left(\\dot{l}^{(1)}_\\bot(s,r) -\\dot{l}^{(1)}_{||}(s,r)\\right)\\left(K-\\frac{\\ell (\\ell +1)}{2r^2}\\right),\n \\\\\n T_3(t,r)\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\\left(\\dot{l}^{(1)}_\\bot (s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\\frac{1-Kr^2}{r}.\n\\end{eqnarray}\n\n\\section{Angular power spectrum and angular growth rate}\\label{sec3}\nIn the previous section, we derived the growing solutions for density contrasts. \nOnce we have specified the isotropic linear perturbations \n$l^{(1)}_{||}$, $l^{(1)}_{\\bot }$, $\\Delta ^{(1)}$ and \nthe initial anisotropic inhomogeneities \n$\\delta^{(\\rm i)+}_{\\ell m}$, $B^{(\\rm i)+}_{\\ell m}$, \nwe obtain the density contrasts $\\delta_{\\ell m}$ by Eq.~\\eqref{dens3}. \nIn this section, \nwe derive the explicit form of the angular power spectrum of \nthe density perturbation $\\delta_{\\ell m}$ \nfor a given set of initial conditions in terms of the standard power spectrum. \nThen, we define the angular growth rate by using the angular power spectrum. \nHereafter, in order to determine the perturbations of order $\\kappa$,\nwe refer to the non-Copernican LTB universe model with uniform big-bang time (see Appendix A).\n\n\n\\subsection{Initial power spectrum of density contrast} \nBy virtue of the uniform big-bang time (see Appendix A),\nthe present non-Copernican LTB universe \napproaches the homogeneous and isotropic universe as time goes back. \nHence, it is reasonable to assume that the initial conditions for the anisotropic \nperturbations are the same as in the case of the FLRW universe.\nThen, the initial power spectrum of the density contrast can be \nexpressed as follows: \n\\begin{eqnarray}\n \\langle\\delta ^{(1)*}(t_{\\rm i},{\\bf k})\\delta ^{(1)}(t_{\\rm i},{\\bf k}')\\rangle\n =(2\\pi)^3\\delta_{\\rm D}^3({\\bf k}-{\\bf k}')P(t_{\\rm i},k),\n \\label{pow1}\n\\end{eqnarray}\nwhere $\\delta_{\\rm D}$ is the Dirac's delta function, \n$t_{\\rm i}$ represents some sufficiently early time already introduced in Eq.~\\eqref{dens2},\nand the Fourier transform of the density contrast is defined by\n\\begin{eqnarray}\n \\delta^{(1)} (t,{\\bf k})=\\int d^3x \\delta^{(1)} (t,{\\bf x})e^{-i{\\bf k\\cdot x}}.\n\\end{eqnarray}\nIf we choose the initial time after recombination, \nthe matter power spectrum including baryons and cold dark matter can be written as\n\\begin{eqnarray}\n P(t_{\\rm i},k) &=& [D^+(t_{\\rm i})]^2P(k), \\nonumber \\\\\n P(k)&=&A_0k^nT^2(k),\n \\label{pow2}\n\\end{eqnarray}\nwhere $A_0$ is a positive constant which represents the amplitude for perturbations on large scales, \n$n$ is constant, and $T(k)$ is the matter transfer function.\nIn this paper, we assume the Harrison-Zel'dovich spectrum $n=1$. \nAs for the transfer function,\nwe adopt the fitting formula developed by Eisenstein \\& Hu~\\cite{Eisenstein:1997ik} \n(see Appendix B). \n\n\\subsection{Angular power spectrum and angular growth rate}\nIn order to observationally study the evolution of perturbations in the non-Copernican universe,\nwe need to specify the observable quantities by using the density contrast \\eqref{dens3}.\nThe simplest quantity that we can currently calculate is the angular power spectrum.\nWe define the angular power spectrum of the density contrast by \n\\begin{eqnarray}\n C_\\ell (t,r)=\n \\frac{r^2}{2\\ell +1}\\sum_{m=-\\ell}^\\ell \\langle\\delta^{*}_{\\ell m}(t,r)\\delta_{\\ell m}(t,r)\\rangle,\n \\label{CL1}\n\\end{eqnarray}\nwhere $*$ denotes the complex conjugate.\n\nHereafter, we focus on the non-spherical perturbations whose wavelengths \nare much smaller than the spatial curvature radius $(k\\gg \\sqrt{|K|})$. \nThen, from Eq. \\eqref{eps4}, we have\n\\begin{eqnarray}\n k^2\\dot{B}^{(1)}(t,{\\bf k})\\simeq 2\\dot{\\delta}^{(1)}(t,{\\bf k}). \n \\label{fou}\n\\end{eqnarray}\nThe initial values $\\delta^{(\\rm i)}_{\\ell m}(r)$ and $B^{(\\rm i)}_{\\ell m}(r)$ \nwhich appear in Eqs.~\\eqref{B1} and \\eqref{dens3} \ncan be written using the Fourier transform of the initial density contrast $\\delta^{(1)}(t_{\\rm i},{\\bf k})$ as\n\\begin{eqnarray}\n \\delta^{(\\rm i)+}_{\\ell m}(r)\n &=&\n \\Big[\\frac{1}{D^+(t_{\\rm i})}\\Big](4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\delta^{(1)} (t_{\\rm i},{\\bf k})j_\\ell (kr)Y^*_{\\ell m}\n ({\\bf \\Omega}_{\\rm k}),\n \\label{de2}\n \\\\\n B^{(\\rm i)+}_{\\ell m}(r)\n &=&\n \\Big[\\frac{1}{D^+(t_{\\rm i})}\\Big](4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\delta^{(1)} (t_{\\rm i},{\\bf k})j_\\ell (kr)\n Y^*_{\\ell m}({\\bf \\Omega}_{\\rm k})\\left(\\frac{2}{k^2}\\right),\n \\label{B2}\n\\end{eqnarray}\nwhere ${\\bf \\Omega}_{\\rm k}$ denotes the polar and azimuthal angles in the Fourier space, and \nwe have used the relation between the expansion coefficient with respect to \n$Y_{\\ell m}({\\bf \\Omega}_{\\rm k})$ and the Fourier transform\n\\begin{eqnarray}\n \\phi_{\\ell m}(t,r)\n =\n (4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\phi (t,{\\bf k})j_\\ell (kr)Y^*_{\\ell m}({\\bf \\Omega}_{\\rm k}),\n \\label{phi2}\n\\end{eqnarray}\nand Eq.~\\eqref{fou}. \nUsing Eqs.~\\eqref{pow1}, \\eqref{pow2}, \\eqref{de2} and \\eqref{B2}, \nthe angular power spectrum of the density contrast \\eqref{dens3} \ncan be rewritten in the following form: \n\\begin{eqnarray}\nC_\\ell (t,r)&=&\n\\epsilon^2{D^+}^2(t)K_1(\\ell ,r)+\\kappa \\epsilon^2 \\Big[2D^+(t)T_1(t,r)K_1(\\ell ,r)\n\\nonumber \\\\\n& &\n+2D^+(t)\\tilde{T}_{2}(t,r,\\ell )K_2(\\ell ,r)+2D^+(t)\\tilde{T}_{3}(t,r)K_3(\\ell ,r)\n\\Big]+{\\cal O}(\\kappa^2\\epsilon^2),\n\\label{CL4}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n K_1(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell ^2(kr),\n \\nonumber \\\\\n K_2(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell ^2(kr)\\left(\\frac{2}{k^2}\\right),\n \\nonumber \\\\\n K_3(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell (kr)j_\\ell ^{'}(kr)\\left(\\frac{2}{k^2}\\right),\n\\end{eqnarray}\nand $\\tilde{T}_2$ and $\\tilde{T}_3$ are defined \nas the short wavelength approximation $(k\\gg \\sqrt{|K|})$ of \n$T_2$ and $T_3$ by \n\\begin{eqnarray}\n \\tilde{T}_2(t,r,\\ell )\n &=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\n \\left(\\dot{l}^{(1)}_\\bot(s,r) -\\dot{l}^{(1)}_{\\parallel}(s,r)\\right)\\left(-\\frac{\\ell (\\ell +1)}{2r^2}\\right),\n \\\\\n \\tilde{T}_3(t,r)\n &=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\\left(\\dot{l}^{(1)}_\\bot (s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\\left(\\frac{1}{r}\\right).\n\\end{eqnarray} \nOnce the initial density power spectrum $P(t_{\\rm i},k)$ is specified, \nwe can calculate the angular power spectrum $C_\\ell (t,r)$ by using Eq.~\\eqref{CL4}.\n\nTo investigate the growth rates of the perturbations, \nwe define the angular growing factor by \n\\begin{eqnarray}\n D_\\ell(t,r)=\\left[\\frac{C_\\ell(t,r)}{C_\\ell(t_{\\rm i},r)}\\right]^{1\/2}. \n \\label{eff1}\n\\end{eqnarray}\nIt is easy to see that the angular growing factor $D_\\ell(t,r)$ is equal to $D^+(t)\/D^+(t_{\\rm i})$ \nup to order $\\epsilon$. \n\nBasically, an observer can see the cosmological structures on his\/her past light cone \nby observations through electromagnetic \nradiation\\footnote{The past light cone of an observer at the event $p$ \nis defined by the boundary of the causal past of $p$, which is usually \ndenoted by $\\dot{J}^-(p)$ in general relativity. \nStrictly speaking, the observer can see the inside of the light cone \nthrough a congruence of the light rays which have experienced caustics caused by gravitational \nlens effects or scattering due to electromagnetic interactions in the real universe.}\nand hence, in the case of the non-Copernican universe model, \nit is useful to consider quantities on the light cone of an observer who stays \nat the symmetry center of the void at present. \nHereafter, for simplicity, we call the observer who stays at the symmetry center of \nthe void at present ``the central observer\", and the past light cone of the central observer \nis denoted by $\\Sigma_{\\rm lc}$. \nThe past light cone $\\Sigma_{\\rm lc}$ is generated by the past-directed outgoing \nradial null geodesics $k^\\mu=(dt\/d\\lambda,dr\/d\\lambda,0,0)$, where $\\lambda$ is \nthe affine parameter. The cosmological redshift $z$ is defined by\n\\begin{equation}\nz=\\frac{dt\/d\\lambda}{(dt\/d\\lambda)_0}-1,\n\\end{equation}\nwhere $dt\/d\\lambda$ and $(dt\/d\\lambda)_0$ are the value at the time of the emission of \na photon and that at the time of the detection of the photon by the central observer, respectively. \nBy using the cosmological redshift $z$ instead of \nthe affine parameter $\\lambda$, the geodesic equations for the generator of the \npast light cone $\\Sigma_{\\rm lc}$ up to order $\\kappa$ are given by\n\\begin{eqnarray}\n\\frac{dr}{dz}&=&\\frac{\\sqrt{1-Kr^2}}{(1+z)aH}\\left[1-\\frac{\\kappa}{2}\n\\left(l_\\parallel+\\frac{1}{H}\\dot{l}_\\parallel\\right)\\right], \\\\\n\\frac{dt}{dz}&=&-\\frac{1}{(1+z)H}\\left(1-\\frac{\\kappa}{2H}\\dot{l}_\\parallel\\right). \n\\end{eqnarray}\nWe denote the solution of the above equations by $t=t_{\\rm lc}(z)$ and $r=r_{\\rm lc}(z)$. \n\nThen, using the angular growing factor, we define the \nangular growth rate on the past light cone $\\Sigma_{\\rm lc}$ as \na function of redshift $z$ as follows:\n\\begin{eqnarray}\n f_\\ell(z)=- \\frac{d [\\ln D_\\ell(t_{\\rm lc}(z),r_{\\rm lc}(z))]}{d \\ln(1+z)}. \n \\label{eff2}\n\\end{eqnarray}\nHere, we note that the angular growing factor and the angular growth rate \ndo not depend on the amplitude $A_0$ in Eq.~\\eqref{pow2}. \nWe also note that the angular growth rate up to order $\\epsilon$ agrees with the growth rate \nusually used in the linear perturbation theory of the FLRW universe, \n $d ({\\rm ln} D^+)\/d ({\\rm ln} a)$. \n\n\n\\section{evolution of density perturbations in the Clarkson-Regis model}\\label{sec4}\n\\subsection{Linearized Clarkson-Regis model}\nIn order to determine the perturbations of order $\\kappa$, \nwe use the non-Copernican LTB universe model \ngiven by Clarkson and Regis~\\cite{Clarkson:2010ej} \n(see Appendix A), which we call the Clarkson and Regis~(CR) model. \nWe shall study the evolution of linear anisotropic perturbations \nin the CR model by using the second order perturbation theory of the FLRW universe filled with dust. \nIn order to approximate the CR model by the linearly perturbed FLRW \nuniverse filled with dust, we must first specify the background FLRW universe. \nHere, we determine the background FLRW universe so that \nthe cosmological density parameter of the background is equal to \n$0.242$, which is equal to the value of the density parameter function $\\Omega_{\\rm M}(r)$\nat the symmetry center of the CR model (see Eq.~\\eqref{CR1}).\n\nWe define the ``density contrast'' of the CR model as\n\\begin{eqnarray}\n & &\n \\Delta^{(\\rm CR)}(t,r)= \\frac{\\rho^{(\\rm CR)}(t,r)-\\rho^{\\rm (CR)}(t,0)}{\\rho^{\\rm (CR)}(t,0)},\n \\label{Del2}\n\\end{eqnarray}\nwhere $\\rho^{(\\rm CR)}$ is the energy density of \nthe CR model. The density contrasts $\\Delta^{(\\rm CR)}$'s on three constant time \nhypersurfaces are depicted in Fig.~\\ref{density_timeconst} as functions of $r$. \n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{density_timeconst.eps}\n \\end{center}\n \\caption{\nDensity contrasts on the spacelike hypersurfaces for\n$t=t_{\\rm lc}(100)$, $t=t_{\\rm lc}(1)$ \nand $t=t_0$, as functions of $r$.\n}\n \\label{density_timeconst}\n\\end{figure}\nHere, we have used the cosmological redshift $z$ to specify each constant \ntime hypersurface given by $t=t_{\\rm lc}(z)$.\nWe can see that the void structure grows with time. \nSince the big-bang time is uniform, there is only the growing mode in the CR model. \nThe void size is about $12{\\rm Gpc}$, and the vicinity of the center \nis locally the dust filled FLRW model \nwith the cosmological density parameter $\\Omega_{\\rm M}=0.242$, whereas \nthe asymptotic region is almost the same as the dust filled FLRW model with $\\Omega_{\\rm M}=0.7$.\nThe Hubble parameter at the center is $H_{0}=74{\\rm kms^{-1}Mpc^{-1}}$. \n\nBy using the density contrast $\\Delta^{(\\rm CR)}$, \nwe give the initial conditions for the isotropic linear density contrast \n$\\Delta ^{(\\rm i)}_{\\pm }$ in Eq.~\\eqref{Del1} as follows.\nAs mentioned, since the CR model has only the growing mode, \nwe should set $\\Delta ^{(\\rm i)}_{-}(r)=0$. By contrast, \n$\\Delta_+^{({\\rm i})}(r)$ is determined by the assumption that \nthe density contrast $\\Delta^{(1)}$ exactly agrees \nwith that of the CR model at the initial time $\\Delta^{\\rm (CR)}(t_{\\rm i},r)$, i.e.,\n\\begin{eqnarray}\n \\Delta ^{(\\rm i)}_+(r)=\\frac{\\Delta^{(\\rm CR)}(t_{\\rm i},r)}{D^+(t_{\\rm i})},\n\\end{eqnarray}\nwhere the initial time is determined by $t_{\\rm i}=t_{\\rm lc}(1000)$. \nOnce the initial condition for the density contrast is given, \nwe obtain all perturbations of order $\\kappa$ \nby solving the perturbation equations up to the corresponding order, and as a result, \nthe linearized CR model is obtained. \n\nTo evaluate the accuracy of the linear approximation,\nwe plot $\\Delta^{(\\rm CR)}$ and $\\Delta^{(1)}$ on the past light cone $\\Sigma_{\\rm lc}$ \nin Fig.~\\ref{density_redshift}.\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{density_redshift.eps}\n \\end{center}\n \\caption{\n Exact density contrast $\\Delta^{(\\rm CR)}$ and \n that of the linearized CR model $\\Delta^{(1)}$ on the past light cone $\\Sigma_{\\rm lc}$,\n plotted as functions of the redshift $z$.\n }\n \\label{density_redshift}\n\\end{figure}\nThe relative error between the exact and linearized CR models is less than $30\\%$\non the light cone $\\Sigma_{\\rm lc}$. \nInside the past light cone $\\Sigma_{\\rm lc}$, \nthe error is smaller than that on the past light cone $\\Sigma_{\\rm lc}$,\nsince the CR model has only the growing mode. \nThere is no qualitative difference between the exact and the linearized CR models,\nand thus we may see the qualitative behavior of linear perturbations in the CR model by \nthe perturbative analysis of the FLRW universe based on the linearized CR model. \n\n\\subsection{Evolution of density contrasts in the CR model}\nLet us consider the evolution of the anisotropic density contrasts in the CR model. \nBy using the angular power spectrum $C_\\ell(t,r)$ given by Eq.~\\eqref{CL4} and \nthe transfer function $T(k)$ given in appendix~\\ref{appendixB},\nwe depict the angular growing factors $D_\\ell(t,r)$'s defined by Eq.~\\eqref{eff1} \nat each comoving distance as functions of $t$ in Fig.~\\ref{growthfactor_CR}. \n\n\\begin{figure}[htbp] \n \\begin{center}\n\\includegraphics[width=10cm,clip]{growthfactor_CR.eps}\n \\end{center}\n \\caption{\nAngular growing factors $D_\\ell$'s in the CR model at \n$r=40{\\rm Gpc}$ (dotted line), $r=4{\\rm Gpc}$ (dashed line) and $r=0$ (dot-dashed line)\ndepicted as functions of $t$. \nThe present time is $H_0t_0=0.83$. \nWe choose $\\ell$ so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$.\n}\n \\label{growthfactor_CR}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{growthfactor_FLRW.eps}\n \\end{center}\n \\caption{\nAngular growing factors $D_\\ell$'s in the dust filled FLRW universe models \nwith $\\Omega_{\\rm M}=0.242$ and $\\Omega_{\\rm M}=0.7$, together with that \nfor the CR model. \n}\n \\label{growthfactor_FLRW}\n\\end{figure}\n\nHere, we introduce a useful quantity defined by \n\\begin{equation}\n{\\tilde k}:=\\frac{\\ell}{r}~.\n\\end{equation} \nNote that $\\tilde k$ is equal to the comoving wave number of the mode \n$\\ell$ at a distance $r$ in the flat sky approximation \n(see e.g., Ref.~\\cite{Lyth:textbook}).\nIn Fig.~\\ref{growthfactor_CR}, the present time is $H_0t_0=0.83$, and $\\ell$ is \nchosen so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$. \nThis choice of $\\ell$ shows us the evolution of perturbations with the size of a cluster of galaxies, i.e., \n$2\\pi \/\\tilde{k} \\sim 10 {\\rm Mpc}$.\n\nWe can see from Fig.~\\ref{growthfactor_CR} that \nthe larger the comoving distance of a perturbation from the symmetry center, \nthe faster the growth of the perturbation. \nThis result may be explained by the fact that the energy density of the CR model \nis a monotonically increasing function of $r$, since the growth rates of \nperturbations in the FLRW universe is a monotonically increasing function of $\\Omega_{\\rm M}$. \nWe also depict $D_\\ell$ of the FLRW universe models \nwith $\\Omega_{\\rm M}=0.242$ and $0.7$, respectively, \ntogether with that of the CR model in Fig.~\\ref{growthfactor_FLRW}. \nWe can see from this figure that $D_\\ell$ of the FLRW universe with \n$\\Omega_{\\rm M}=0.242$ agrees with $D_\\ell(t,r=0)$ of the CR model. \nWe note that $D_\\ell$ in the FLRW universe with $\\Omega_{\\rm M}=0.7$ \ndoes not agree with that far from the void ($r=40{\\rm Gpc}$) in the CR model.\nThis result might not be real, but rather could be an error caused by using \nthe linearized CR model, since the error due to the linear\napproximation becomes larger for larger $r$. \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[trim = 5 110 5 110 ,width=14cm,clip]{kdepend.eps}\n \\end{center}\n \\caption{\n Angular growing factors $D_\\ell(t,r=4{\\rm Gpc})$ in the CR model for\n $\\tilde{k}=0.01{\\rm Mpc^{-1}}$ (solid line), $0.1{\\rm Mpc^{-1}}$ (dashed line) \n and $1{\\rm Mpc^{-1}}$ (dot-dashed line), depicted \n as functions of $t$.\n The right panel shows a close-up of the $0.7 0$, \n\tthen $t$ is a product that maximizes the quasi-linear utility of agent \n\t$s$, and the utility for agent $s$ is non-negative. That is, \n\t\\begin{equation}\n\t\t\\label{eq:envyfree}\n\t\t\\forall s, t : x^{st} > 0 \\Rightarrow \n\t\tw^{st} - p^t = \\max_k \\{w^{sk} - p^k\\} \\geq 0 \\enspace.\n\t\\end{equation}\n\\end{definition}\n\n\\begin{definition}\n\tAn allocation $\\bm{x}$ is {\\em market-clearing}\n\tif all demand constraints and supply constraints hold with equality\n\tThat is, \n\t$$\\forall 1 \\leq s \\leq \\ell: \\sum_{t=1}^m x^{st} = \\alpha^s ~,~\n\t\\forall 1 \\leq t \\leq m: \\sum_{s=1}^\\ell x^{st} = \\beta^t \\enspace.$$\n\\end{definition}\n\nThe social welfare maximization problem for a fractional assignment problem\nis characterized by the following linear program (P) and its dual (D).\n\\begin{align*}\n\t\\text{{\\bf (P)} Maximize} & ~ ~ \\Sigma_{s = 1}^\\ell \n\t\\Sigma_{t = 1}^m x^{st} w^{st} & & \\text{s.t.} &\n\t\\text{{\\bf (D)} Minimize} & ~ ~ \\Sigma_{s = 1}^\\ell \\alpha^s u^s + \n\t\\Sigma_{t = 1}^m \\beta^t p^t & & \\text{s.t.} \\\\\n\t\\Sigma_{t = 1}^m x^{st} & \\leq \\alpha^s & & \\forall s & \n\tu^s + p^t & \\geq w^{st} & & \\forall s, t \\\\\n\t\\Sigma_{s = 1}^\\ell x^{st} & \\leq \\beta^t & & \\forall t &\n\tu^s & \\geq 0 & & \\forall s \\\\\n\tx^{st} & \\geq 0 & & \\forall s, t &\n\tp^t & \\geq 0 & & \\forall t\n\\end{align*}\n\n\\begin{align*}\n\\end{align*}\n\nWe will introduce two useful lemmas about the connection between \nenvy-free prices and social welfare maximization for fractional\nassignment problems. These lemmas were known in different forms in the\neconomics literature \\cite{gul1999walrasian}.\n\n\\begin{lemma}\n\t\\label{lemma:assignment1}\n\tIf there exist envy-free prices $\\bm{p}$ for a market-clearing\n\tallocation $\\bm{x}$, then $\\bm{x}$ maximizes the social welfare,\n\tthat is, $\\bm{x} \\cdot \\bm{w} = \\max_{\\bm{z}} \\bm{z} \\cdot \\bm{w}$.\n\\end{lemma}\n\n\\begin{proof}\n\tSuppose there exist envy-free prices $\\bm{p}$ for \n\tan allocation $\\bm{x}$. Let $u^s = \\max_t \\left\\{w^{st}-p^t\\right\\}$. \n\tWe have that $u^s + p^t \\geq w^{st}$ for all $s, t$. So \n\t$(\\bm{u}, \\bm{p})$ is a feasible solution for the dual LP.\n\n\tMoreover, by definition of envy-freeness, we have\n\t$$\\forall s, t : x^{st} > 0 \\Rightarrow u^s = w^{st} - p^t \\enspace.$$\n\t\n\tTherefore, we get that\n\t$$\\sum_{s=1}^\\ell \\sum_{t=1}^m x^{st} w^{st} = \n\t\\sum_{s=1}^\\ell \\sum_{t=1}^m x^{st} (u^s + p^t) \n\t= \\sum_{s=1}^\\ell \\alpha^s u^s + \\sum_t \\beta^t p^t \\enspace.$$\n\n\tThe last equality holds because $\\bm{x}$ is market clearing.\n\tNotice that $\\bm{x}$ is a feasible solution to the primal LP. \n\tBy duality theorem, we get that the allocation $\\bm{x}$ maximizes \n\tthe social welfare for the fractional assignment problem.\n\\end{proof}\n \n\\begin{lemma}\n\t\\label{lemma:assignment2}\n\tIf an allocation $\\bm{x}$ maximizes the social welfare, then there exist\n\tenvy-free prices $\\bm{p}$ for the fractional assignment problem.\n\\end{lemma}\n\n\\begin{proof}\n\tSuppose the allocation $\\bm{x}$ maximizes the social\n\twelfare. Let $(\\bm{u}, \\bm{p})$ be an optimal solution to the dual\n\tLP. By complementary slackness we get that $x^{st} > 0$ only if the\n\tcorresponding dual constraint is tight, that is, $u^s + p^t = w^{st}$.\n\tTherefore, $x^{st} > 0$ implies that $w^{st} - p^t = u^s \\geq \n\tw^{sk} - p^k$ for all $k$. Thus $\\bm{p}$ is a collection of envy-free \n\tprices for the allocation $\\bm{x}$ in this fractional assignment problem.\n\\end{proof}\n\nNote that the above proof also gives a poly-time algorithm for\nfinding the welfare maximizing allocation $\\bm{x}$ and the corresponding\nenvy-free prices $\\bm{p}$ by solving the primal and dual LPs.\nMoreover, we also get that the envy-free prices $\\bm{p}$ satisfy a weak \nuniqueness in the sense that it must be part of an optimal solution for the \ndual LP.\n\n\\begin{corollary}\n\tThere exists a poly-time algorithm that computes \n\tthe welfare-maximizing market-clearing allocation and \n\tthe envy-free prices. \n\\end{corollary}\n\n\\subsection{Characterizing BIC via envy-free prices.}\n\nWe first introduce some notations that will simplify our discussion.\nGiven a mechanism $\\mathcal{M}$ for a multi-parameter mechanism design problem\n$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$, we will consider the\nexpected values and expected prices for each agent when it choose a specific \nstrategy (each strategy corresponds to reporting a specific valuation).\n\nAssuming the other agents report their valuations truthfully, \nagent $i$'s expected value of the service it gets, when the \ngenuine valuation is $v^s_i$ and the reported valuation is $v^t_i$, is\n$$w^{st}_i = \\e{\\bm{v}_{-i},(\\bm{S},\\bm{p})\\sim\\mathcal{M}(v^t_i,\\bm{v}_{-i})}{\nv^s_i(S_i)} \\enspace.$$\n\nSimilarly, we let $p^i_t$ denote the expected price the mechanism would charge\nto agent $i$ if its reported valuation is $v^t_i$, that is, \n$$p^t_i = \\e{\\bm{v}_{-i},(\\bm{S},\\bm{p})\\sim\\mathcal{M}(v^t_i,\\bm{v}_{-i})}{p_i} \n\\enspace.$$\n\nBy the definition of BIC and IR, the mechanism $\\mathcal{M}$ is BIC and IR if and \nonly if for any $1 \\leq i \\leq n$ and $1 \\leq s \\leq \\ell$,\n\\begin{equation}\n\tw^{ss}_i - p^s_i = \\max_t \\{w^{st}_i - p^t_i \\} \\geq 0\n\t\\enspace. \\label{eq:bic}\n\\end{equation}\n\nThe above equation \\eqref{eq:bic} is similar to \nequation \\eqref{eq:envyfree} in the definition of envy-freeness in fractional\nassignment problem. In fact,\nthe key observation is that the above BIC condition is equivalent to the \nenvy-free condition for a set of properly chosen fractional assignment \nproblems.\n\n\\paragraph{Induced assignment problems.}\n\nFor each agent $i$, we will consider the following {\\em induced assignment \nproblem}. \nWe consider $\\ell$ virtual buyers with demands $f_i(1), \\dots, f_i(\\ell)$ \nrespectively, and $\\ell$ virtual products with supplies $f_i(1), \\dots, \nf_i(\\ell)$ respectively. For each virtual buyer $s$ and each virtual product\n$t$, let virtual buyer $s$ has value $w^{st}_i$ on virtual product $t$.\nWe will refer to this fractional assignment problem the {\\em induced\nassignment problem} of agent $i$.\n\n\\medskip\n\nLet us consider the {\\em identity allocation} $\\bm{x}_i$ defined as follows:\n$$x^{st}_i = \\left\\{\\begin{aligned}\n\t& f_i(s) & & \\text{, if } s = t \\enspace,\\\\\n\t& 0 & & \\text{, otherwise.}\n\\end{aligned}\\right.$$ \n\nWe can easily verify that \na collection of prices $\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ satisfies\nconstraint \\eqref{eq:bic} if and only if $\\bm{p}_i$ satisfies the envy-free\ncondition \\eqref{eq:envyfree} of the induced assignment problem of agent $i$\nwith respect to the above identity allocation.\nHence, we have the following connection between BIC mechanism and the\nenvy-free prices of the induced assignment problems.\n\n\\begin{lemma}[Characterization Lemma~\\cite{rochet1987necessary}]\n\t\\label{lemma:characterization}\n\tA mechanism $\\mathcal{M}$ is BIC if and only if in the induced assignment \n\tproblem of each agent $i$ the identity \n\tallocation $\\bm{x}_i = \\{x^{st}_i\\}_{1 \\leq s,t \\leq \\ell}$ \n\tmaximizes the social welfare, and $\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ \n\tare chosen to be the corresponding envy-free prices.\n\\end{lemma}\n\n\\paragraph{Comparing with Myerson's characterization.}\n\nSuppose the problem falls into the single-parameter domain. Each\nvaluation $v^s_i$ is represented by a single non-negative real number. With\na little abuse of notation, we let $v^s_i$ denote this value. Without loss\nof generality, we assume that $v^1_i > \\dots > v^\\ell_i$. We let\n$y^t_i$ denote the probability that agent $i$ would be served if the reported\nvalue was $v^t_i$. The values $\\bm{w}_i$ in the fractional assignment\nproblems of agent $i$ are $w^{st}_i = v^s_i y^t_i$ for $1 \\leq s, t \\leq \\ell$.\nMyerson's famous characterization \\cite{myerson1981optimal}\nof truthfulness in single-parameter domain\nimplied that the mechanism is BIC if and only if $y^1_i \\geq\n\\dots \\geq y^\\ell_i$. Indeed, due to rearrangement inequality, the identity \nallocation $\\bm{x}_i$ maximizes the social welfare if and only if \n$y^1_i \\geq \\dots \\geq y^\\ell_i$. Thus, the characterization lemma implies\nMyerson's characterization in the single-parameter domain.\n\n\\section{Reduction for social welfare}\n\\label{sec:socialwelfare}\n\nLemma \\ref{lemma:characterization} suggests an interesting connection between \nBIC and envy-free prices for the induced assignment problems. \nHence, given an algorithm $\\mathcal{A}$, we will manipulate the allocation by \n$\\mathcal{A}$ based on this connection in order to make \nit satisfy the condition in Lemma \\ref{lemma:characterization}.\n\n\\subsection{Main ideas.}\n\nLet us first briefly convey two key ideas in the construction of the\nwelfare-preserving reduction.\n\nThe first idea is to decouple the reported agent valuations and the \ninput agent valuations for algorithm $\\mathcal{A}$. More\nconcretely, we will introduce $n$ intermediate algorithm $\\mathcal{B}_1, \\dots,\n\\mathcal{B}_n$. Each $\\mathcal{B}_i$ will take the reported valuation $v'_i$ as\ninput, then output a valuation $\\widetilde{v}_i$ that may or may not equals \n$v'_i$. Then, we will use algorithm $\\mathcal{A}$ to compute the allocation $\\bm{S}$\nfor agent valuations $\\widetilde{v}_1, \\dots, \\widetilde{v}_n$, \nand allocate services according to $\\bm{S}$. \n\nIf we revisit the values \n$\\widetilde{\\bm{w}}_i$ in the induced assignment problem of agent $i$ after\nthis manipulation, we will get that for any $1 \\leq s, t \\leq \\ell$,\n$$\\widetilde{w}^{st}_i = \\e{\\bm{v}_{-i}, \\widetilde{\\bm{v}} \\sim \n\\mathcal{B}(v^t_i, \\bm{v}_{-i}), \\bm{S} \\sim \\mathcal{A}(\\widetilde{\\bm{v}})}\n{v^s_i(S_i)} \\enspace.$$\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.45\\textwidth]{reduction.pdf}\n\t\\caption{High-level picture of the reduction for social welfare.\n\t$\\mathcal{B}_i$'s are intermediate algorithms for manipulating the \n\tinput of algorithm $\\mathcal{A}$.\n\t$\\widetilde{v}_i$'s are the reported valuations. $v'_i$'s are the \n\tmanipulated input valuations for algorithm $\\mathcal{A}$. $\\bm{S}$ is\n\tthe final allocation.}\n\t\\label{fig:reduction}\n\\end{figure}\n\nBy Lemma \\ref{lemma:characterization}, we need to choose $\\mathcal{B}_i$'s carefully, \nso that the identity allocations in the manipulated assignment problems are\nwelfare-maximizing allocations. However, from the above equation we can see \nthat by using $\\mathcal{B}_i$ to manipulate the $i^{th}$ valuation, we may change\nnot only the structure of the induced assignment problem of agent $i$, but the\nstructure of the induced assignment problems of other agents as well. Hence,\nwe need to handle such correlation among the induced assignment problems when\nwe choose intermediate algorithms $\\mathcal{B}_1, \\dots, \\mathcal{B}_n$.\n\nThe idea that handles this correlation is to impose an extra constraint on\neach intermediate algorithm $\\mathcal{B}_i$: if the input valuation $v'_i$ is \ndrawn from the distribution $F_i$, then the output valuation \n$\\widetilde{v}_i$ also follows the same distribution, that is, \nfor all $1 \\leq i \\leq n$ and $1 \\leq t \\leq \\ell$,\n\\begin{equation}\n\t\\label{eq:interconstraint}\n\t\\pr{v'_i \\sim F_i, \\widetilde{v}_i \\sim \\mathcal{B}_i(v'_i)}{\\widetilde{v}_i =\n\tv^t_i} = f_i(t) \\enspace.\n\\end{equation}\n\t\nWith this extra constraint, the values $\\widetilde{\\bm{w}}_i$ after the \nmanipulation in the induced assignment problem of agent $i$ becomes\n\\begin{eqnarray*}\n\t\\widetilde{w}^{st}_i & = & \\e{\\bm{v}_{-i} \\sim F_{-i}, \n\t\\widetilde{\\bm{v}}\\sim\\mathcal{B}(v^t_i, \\bm{v}_{-i}), \\bm{S} \\sim \n\t\\mathcal{A}(\\widetilde{\\bm{v}})}{v^s_i(S_i)} \\\\\n\t& = & \\e{\\widetilde{\\bm{v}}_{-i} \\sim F_{-i}, \\widetilde{v}_i \\sim \n\t\\mathcal{B}_i(v^t_i), \\bm{S} \\sim \\mathcal{A}(\\widetilde{\\bm{v}})}{v^s_i(S_i)} \\\\\n\t& = & \\e{\\bm{v}_{-i} \\sim F_{-i}, \\widetilde{v}_i \\sim \n\t\\mathcal{B}_i(v^t_i), \\bm{S} \\sim \\mathcal{A}(\\widetilde{v}_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i)} \\enspace.\n\\end{eqnarray*}\n\nThus, from the Bayesian viewpoint of agent $i$, the intermediate algorithms\n$\\mathcal{B}_{-i}$ of other agents are transparent. This property enables us to \nmanipulate the structure of each assignment problem separately.\n\n\\subsection{Black-box reduction.}\n\nGiven an algorithm $\\mathcal{A}$, the black-box reduction for social welfare will \nconvert algorithm $\\mathcal{A}$ into the following mechanism $\\mathcal{M}_\\mathcal{A}$:\n\n\\begin{enumerate}\n\t\\item For each agent $i$, $1 \\leq i \\leq n$ \n\t\t\\hspace*{\\fill} {\\bf (Pre-computation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Estimate the values\n\t\t\t\t$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \n\t\t\t\tof the induced assignment problem of $i$ with respect to \n\t\t\t\talgorithm $\\mathcal{A}$. Let \n\t\t\t\t$\\hat{\\bm{w}}_i = \\{\\hat{w}^{st}_i\\}_{1 \\leq s, t\\leq \\ell}$\n\t\t\t\tdenote the estimated values.\n\t\t\t\\item Find the social welfare maximizing allocation \n\t\t\t\t$\\bm{x}_i = \\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$\n\t\t\t\tand the corresponding envy-free prices \n\t\t\t\t$\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ for the induced \n\t\t\t\tassignment problem of agent $i$ with estimated values.\n\t\t\\end{enumerate}\n\t\\item Manipulate the valuations with intermediate algorithms\n\t\t$\\mathcal{B}_i$, $1 \\leq i \\leq n$, as follows:\n\t\t\\hspace*{\\fill} {\\bf (Decoupling)}\n\t\t\\begin{enumerate}\n\t\t\t\\item[] Suppose the reported valuation of agent $i$ is\n\t\t\t\t$v'_i = v^s_i$, $1 \\leq i \\leq n$.\n\t\t\t\tLet $\\widetilde{v}_i = \\mathcal{B}_i(v'_i) = v^t_i$ \n\t\t\t\twith probability $x^{st}_i \/ f_i(s)$ for $1 \\leq t \\leq \\ell.$\n\t\t\\end{enumerate}\n\t\\item Allocate services according to $\\mathcal{A}(\\widetilde{\\bm{v}})$. \n\t\t\\hspace*{\\fill} {\\bf (Allocation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Let $\\bm{S} = (S_1, \\dots, S_n)$ denote the allocation by\n\t\t\t\talgorithm $\\mathcal{A}$ with input $\\widetilde{\\bm{v}}$.\n\t\t\t\\item For each agent $i$, suppose the reported valuation is \n\t\t\t\t$v'_i = v^s_i$ and the manipulated valuation is \n\t\t\t\t$\\widetilde{v}_i = v^t_i$, charge agent $i$ with price \n\t\t\t\t$p^t_i v^s_i(S_i) \/ \\hat{w}^{st}_i$.\n\t\t\\end{enumerate}\n\\end{enumerate}\n\nThe following theorem\nstates that this reduction produces BIC while preserving the performance\nwith respect to social welfare.\n\n\\begin{theorem}\n\t\\label{thm:welfarereduction}\n\tSuppose $\\mathcal{A}$ is an algorithm for a multi-parameter mechanism design\n\tproblem $\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$.\n\t\\begin{enumerate}\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ are \n\t\t\taccurate, then mechanism $\\mathcal{M}_\\mathcal{A}$ is BIC, IR, and \n\t\t\tguarantees at least $SW^\\mathcal{A}$ of social welfare. \n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that for any \n\t\t\t$1 \\leq s, t \\leq \\ell$, \n\t\t\t$\\hat{w}^{st}_i \\in [(1 - \\epsilon) w^{st}_i, \n\t\t\t(1 + \\epsilon) w^{st}_i]$, \n\t\t\tthen mechanism $\\mathcal{M}_\\mathcal{A}$ is $4 \\epsilon v_{max}$-BIC, IR, \n\t\t\tand guarantees at least $(1 - 2\\epsilon) \\cdot SW^\\mathcal{A}$ of \n\t\t\tsocial welfare.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ \n\t\t\tsatisfy that for any $1 \\leq s, t \\leq \\ell$,\n\t\t\t$\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon, \n\t\t\tw^{st}_i + \\epsilon]$, then mechanism $\\mathcal{M}_\\mathcal{A}$ is \n\t\t\t$4 \\epsilon$-BIC, IR, and guarantees at least \n\t\t\t$SW^\\mathcal{A} - 2 n \\epsilon$ of social welfare.\n\t\\end{enumerate}\n\\end{theorem}\n\nLet us illustrate the proof of part 1. The proofs of the other two parts are \ntedious and simple calculations along the same line. We will omit these proofs\nin this extended abstract.\n\n\\begin{proof}\n\tWe consider the case when the estimated values $\\hat{\\bm{w}}_i$ \n\tare accurate,\n\tthat is, $\\hat{w}^{st}_i = w^{st}_i$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\n\t\\paragraph{Individual rationality.}\n\tBy our choice of envy-free prices, we have that \n\t$p^t_i \\leq w^{st}_i$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\tThus, we always guarantee\n\t$$p^t_i \\, \\frac{v^s_i(S_i)}{w^{st}_i} \\leq v^s_i(S_i) \\enspace.$$ \n\n\tSo the mechanism $\\mathcal{M}_\\mathcal{A}$ that we get from the reduction\n\talways provides non-negative utilities for the agents.\n\tEssentially the same proof also shows IR for part 2 and 3.\t\n\t\n\t\\paragraph{Bayesian incentive compatibility.}\n\tWe will first show that the intermediate algorithms in the decoupling\n\tstep of the reduction satisfy constraint \\eqref{eq:interconstraint}.\n\tLet $\\bm{x}_i$ denote the social welfare maximizing allocation\n\tthat the reduction finds\n\tfor the induced assignment problem of agent $i$ for $1 \\leq i \\leq n$.\n\tNote that these social welfare maximizing allocations are market-clearing. \n\tWe have that if the reported valuation $v'_i$ follows the distribution \n\t$F_i$, then the distribution of the manipulated valuation \n\t$\\widetilde{v}_i$ satisfies that\n\t$$\\pr{}{\\widetilde{v}_i = v^t_i} = \\sum_{s=1}^\\ell \\pr{}{v'_i = v^s_i} \\,\n\t\\pr{}{\\widetilde{v}_i = v^t_i : v'_i = v^s_i} \n\t= \\sum_{s=1}^\\ell f_i(s) \\, \\frac{x^{st}_i}{f_i(s)} \n\t= \\sum_{s=1}^\\ell x^{st}_i = f_i(t) \\enspace.$$\n\n\tIndeed, the intermediate algorithms satisfy constraint\n\t\\eqref{eq:interconstraint}. Thus, for each $1 \\leq i \\leq n$ \n\tthe intermediate algorithm $\\mathcal{B}_i$\n\tonly changes the structure of induced assignment problem of agent $i$\n\tand leaves the induced assignment problems of other agents untouched.\n\t\n\tNext, we will verify that in each of the manipulated assignment \n\tproblem, the identity allocation maximizes the social welfare and the\n\tprices are the corresponding envy-free prices.\n\n\tFor each agent $i$,\n\twe let $\\widetilde{\\bm{w}}_i = \n\t\\{\\widetilde{w}^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$\n\tand $\\widetilde{\\bm{p}}_i = \n\t(\\widetilde{p}^1_i, \\dots, \\widetilde{p}^\\ell_i)$\n\tdenote the values and the expected prices of the virtual products \n\trespectively in\n\tthe manipulated assignment problem of agent $i$. We have that for any \n\t$1 \\leq r, s \\leq \\ell$, \n\t\\begin{align*}\n\t\t\\widetilde{w}^{rs}_i & = \n\t\t\\sum_{t=1}^\\ell \\pr{}{\\widetilde{v}_i = v^t_i} \\e{\\bm{v}_{-i}, \n\t\t\\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^r_i(S_i)} \n\t\t= \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, w^{rt}_i \\enspace, \\\\\n\t\t\\widetilde{p}^s_i \n\t\t& = \\sum_{t=1}^\\ell \\pr{}{\\widetilde{v}_i = v^t_i} \n\t\t\\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{p^t_i \\frac{v^s_i(S_i)}{w^{rs}_i}} \n\t\t= \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, p^t_i\\enspace.\n\t\\end{align*}\n\n\tThus, in the manipulated assignment problem of agent $i$,\n\tthe utility of the virtual buyer $r$ of the virtual product $s$,\n\t$1 \\leq r, s \\leq \\ell$, is \n\t$$\\widetilde{w}^{rs}_i - \\widetilde{p}^s_i = \\sum_{t=1}^\\ell\n\t\\frac{x^{st}_i}{f_i(s)} \\, (w^{rt}_i - p^t_i) \n\t\\leq \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, \\max_k \n\t\\{w^{rk}_i - p^k_i\\} = \\max_k \\{w^{rk}_i - p^k_i\\} \\enspace.$$\n\n\tSince $\\bm{p}_i$ are chosen to be the envy-free prices, we have that\n\t$x^{rt}_i > 0$ only if $w^{rt}_i - p^t_i = \\max_k \\{w^{rk}_i - p^k_i\\}$.\n\tHence, when agent $i$ reports its valuation truthfully, that is, $r = s$,\n\tthe above inequality holds with equality. \n\tSo the $\\widetilde{p}_i$ are envy-free\n\tprices with respect to the identity allocation $\\widetilde{\\bm{x}}_i$\n\tof the manipulated assignment problem of agent $i$. By Lemma \n\t\\ref{lemma:assignment1}\n\twe know the allocation $\\widetilde{\\bm{x}}_i$ maximizes the social welfare.\n\tThus, mechanism $\\mathcal{M}_\\mathcal{A}$ is BIC according to Lemma \n\t\\ref{lemma:characterization}.\n\t\n\t\\paragraph{Social welfare.} The expected social welfare for this \n\tmechanism is $\\sum_{i=1}^n \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell\n\tx^{st}_i w^{st}_i$. Since for any $1 \\leq i \\leq n$\n\tthe allocation $\\bm{x}_i$ maximizes the social welfare for the \n\tinduced assignment problem of agent $i$, the social welfare of $\\bm{x}_i$\n\tis at least as large as that of the identity allocation, that is,\n\t$$\\forall i : \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i w^{st}_i \\geq \n\t\\sum_{s=1}^\\ell f_i(s) w^{ss}_i \n\t= \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}{v_i(S_i)} \n\t\\enspace.$$\n\n\tThus, we have that\n\t$$SW^{\\mathcal{M}_\\mathcal{A}} =\n\t\\sum_{i=1}^n \\sum_{s, t=1}^\\ell x^{st}_i w^{st}_i\n\t\\geq \\sum_{i=1}^n \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}\n\t{v_i(S_i)} = \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}\n\t{\\sum_{i=1}^n v_i(S_i)} = SW^\\mathcal{A} \\enspace.$$\n\\end{proof}\n\n\\subsection{Estimating values by sampling.}\n\nThere is still one technical issue that we need to settle in the reduction.\nIn this section, we will briefly discuss how to use the standard sampling\ntechnique to obtain good estimated values of \n$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \nfor the induced assignment problem of agent $i$ for $1 \\leq i \\leq n$. \n\nBy definition, $w^{st}_i$ is the expectation of a random\nvariable $v^s_i(S_i)$, where $S_i$ is the allocated service given by \n$\\mathcal{A}$ over random realization of the valuations $\\bm{v}_{-i}$ of other agents\nand random coin flips of the algorithm. Hence, if the standard deviation \nof $v^s_i(S_i)$ is not too large compared to its mean (no more than a \npolynomial factor), then we can draw polynomially many independent samples and \ntake the average value as our estimated value. \nMore concretely, the sampling algorithm proceeds as follows.\n\\begin{enumerate}\n\t\\item Draw $N = 4 \\, c^2 \\log(n \\ell^2 \/\\epsilon) \/ \\epsilon^2$ independent \n\t\tsamples of $\\bm{v} \\sim \\bm{F}$ conditioned on that the valuation\n\t\tof agent $i$ is $v^t_i$, where \n\t\t$$c = \\frac{\\sd{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i)}}\n\t\t{\\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t\t\\enspace.$$\n\t\tLet $\\bm{v}^1, \\dots, \\bm{v}^N$ denote these $N$ sample.\n\t\\item Use algorithm $\\mathcal{A}$ to compute an allocation $\\bm{S}^k \\sim \n\t\t\\mathcal{A}(\\bm{v}^k)$ for each sample $\\bm{v}^k$, $1 \\leq k \\leq N$.\n\t\\item Let $\\hat{w}^{st}_i$ be the average of $v^s_i(S^k_i)$, $1 \\leq k\n\t\t\\leq N$.\n\\end{enumerate}\n\n\\begin{lemma}\n\t\\label{lemma:relative}\n\tThe estimated values $\\hat{\\bm{w}}_i$, $1 \\leq i \\leq n$, by the above\n\tsampling procedure satisfy for any $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \n\t\\ell$,\n\t$$\\hat{w}^{st}_i \\in \\left[(1 - \\epsilon) w^{st}_i,\n\t(1 + \\epsilon) w^{st}_i\\right]$$\n\twith probability at least $1 - \\epsilon$.\n\\end{lemma}\n\n\\begin{proof}\n\tWe shall have that \n\t\\begin{align*}\n\t\t\\e{}{\\hat{w}^{st}_i} & = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\t\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i))} = w^{st}_i \\enspace, \\\\\n\t\t\\sd{}{\\hat{w}^{st}_i} & = \\frac{1}{\\sqrt{N}} \\, \\sd{\\bm{v}_{-i}, \\bm{S} \n\t\t\\sim \\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i)} \n\t\t= \\frac{c}{\\sqrt{N}} \\, \\e{}{\\hat{w}^{st}_i}\n\t\t= \\frac{c}{\\sqrt{N}} \\, w^{st}_i \\enspace.\n\t\\end{align*}\n\t\n\tBy Chernoff-Hoeffding bound, we get\n\t\\begin{eqnarray*}\n\t\t\\pr{}{\\left|\\hat{w}^{st}_i - w^{st}_i\\right| > \n\t\t\\epsilon w^{st}_i} \n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t\\frac{\\epsilon \\sqrt{N}}{c} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t2 \\, \\sqrt{\\log{(n \\ell^2\/\\epsilon)}} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& \\leq & e^{- \\log{(n \\ell^2 \/\\epsilon)}} = \\frac{\\epsilon}{n \\ell^2}\n\t\t\\enspace.\n\t\\end{eqnarray*}\n\n\tSince we only need to estimate $n \\ell^2$ values,\n\tby union bound we get that with probability at least $1 - \\epsilon$\n\tthe estimated value $\\hat{w}^{st}_i$ is within\n\t$\\epsilon$ relative error compared to $w^{st}_i$ for all\n\t$1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\\end{proof}\n\nThus, if the allocation algorithm $\\mathcal{A}$ admits $SW^\\mathcal{A}$ social welfare and\nthe ratio $c$ is only polynomially large, then by part 2 of Theorem \n\\ref{thm:welfarereduction} we get that mechanism $\\mathcal{M}_\\mathcal{A}$ gives\n$(1-3\\epsilon) \\cdot SW^\\mathcal{A}$ social welfare and is $4\\epsilon v_{max}$-BIC.\nThe running time is polynomial in the input size and $1 \/ \\epsilon$, assuming\na black-box call to algorithm $\\mathcal{A}$ can be done in a single step. \nIn other words, we get a FPTAS reduction.\n\n\\medskip\n\nThe next lemma gives an alternative bound of the sampling error with respect\nto absolute error.\n\n\\begin{lemma}\n\t\\label{lemma:absolute}\n\tIf we draw $N' = 4\\log(n\\ell^2\/\\epsilon)\/\\epsilon^2$ independent samples, \n\tthen with probability at least $1-\\epsilon$ the estimated values \n\t$\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon v_{max}, \n\tw^{st}_i + \\epsilon v_{max}]$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t\n\t\\leq \\ell$.\n\\end{lemma}\n\n\\begin{proof}\n\tIn this case, we have \n\t\\begin{align*}\n\t\t\\e{}{\\hat{w}^{st}_i} & = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\t\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i))} = w^{st}_i \\enspace, \\\\\n\t\t\\sd{}{\\hat{w}^{st}_i} & = \\frac{1}{\\sqrt{N'}} \\, \\sd{\\bm{v}_{-i}, \n\t\t\\bm{S}\\sim\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i)} \n\t\t\\leq \\frac{1}{\\sqrt{N'}}\\max_{S_i} \\, v^s_i(S_i) \\leq \\frac{1}\n\t\t{\\sqrt{N'}} \\, v_{max}\n\t\t\\enspace.\n\t\\end{align*}\n\t\n\tBy Chernoff bound we get that\n\t\\begin{eqnarray*}\n\t\t\\pr{}{\\left|\\hat{w}^{st}_i - w^{st}_i\\right| > \n\t\t\\epsilon v_{max}}\n\t\t& \\leq & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t\\frac{\\epsilon}{\\sqrt{N'}} \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t2 \\, \\sqrt{\\log{(n \\ell^2\/\\epsilon)}} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& \\leq & e^{- \\log{(n \\ell^2 \/\\epsilon)}} = \\frac{\\epsilon}{n \\ell^2}\n\t\t\\enspace.\n\t\\end{eqnarray*}\n\n\tBy union bound, we have $\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon v_{max},\n\tw^{st}_i + \\epsilon v_{max}]$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t\n\t\\leq \\ell$.\n\\end{proof}\n\nSuppose the ratio $v_{max} \/ SW^\\mathcal{A}$ is upper bounded by a polynomial of \nthe input size.\nThen, if we choose $\\epsilon = \\delta \\, SW^\\mathcal{A} \/ 2n v_{max}$ in the above\nlemma, we will get that\n$$\\left|\\hat{w}^{st}_i - w^{st}_i\\right| < \\delta \\, SW^\\mathcal{A} \/ 2n \\enspace.$$\n\nBy part 3 of Theorem {\\ref{thm:welfarereduction} we obtain that\nmechanism $\\mathcal{M}_\\mathcal{A}$ provides at least $(1 - \\delta) SW^\\mathcal{A}$ of social\nwelfare and is $4 \\epsilon$-BIC and IR.\nThe running time is polynomial in the input size and $1 \/ \\delta$.\n\n\\section{Reductions for revenue and residual surplus}\n\nIn the reduction for social welfare in the previous section, \nwe only consider market-clearing allocations in the induced assignment \nproblems. This is because for any agent \n$i$, we want to make sure that the intermediate algorithm $\\mathcal{B}_i$ \nis transparent to all agents except agent $i$. If we restrict ourselves to \nmarket-clearing allocations, we do not know any way to get reasonable bounds \non revenue and residual surplus.\n\nHowever, if we focus on an important sub-class of multi-parameter mechanism\ndesign problems that includes the combinatorial auction problem and its special\ncases, then we have some flexibility in choosing the allocations for the \ninduced assignment problem and obtain theoretical bounds on revenue and\nresidual surplus. More concretely, we will consider mechanism design problems\nthat are {\\em downward-closed}. We let $\\phi$ denote the null service so that\nallocating $\\phi$ to an agent implies that agent is not served, that is,\n$v_i(\\phi) = 0$ for all $1 \\leq i \\leq n$. \n\n\\begin{definition}\n\tA multi-parameter mechanism design problem \n\t$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$\n\tis {\\em downward-closed} if for any \n\tfeasible allocation $\\bm{S} = (S_1, \\dots, S_n) \\in \\bm{J}$ and any\n\t$1 \\leq i \\leq n$, the allocation \n\t$(S_1, \\dots, S_{i-1}, \\phi, S_{i+1}, \\dots, S_n)$ is also feasible.\n\\end{definition}\n\nWe let $\\delta = \\min \\{f_i(s) : 1 \\leq i \\leq n, 1 \\leq s \\leq \\ell, \nf_i(s) > 0\\}$ denote the granularity of the prior distributions. We will \nprove the following result.\n\n\\begin{theorem}\n\t\\label{thm:otherreduction}\n\tFor any algorithm $\\mathcal{A}$, there is a mechanism that is IR, BIC, and \n\tprovides at least $\\Omega(SW^\\mathcal{A} \/ \\log(1 \/ \\delta))$ of revenue \n\t(residual surplus).\n\\end{theorem}\n\n\\subsection{Meta-reduction.}\n\nWe will first introduce a meta-reduction scheme based on algorithms that\ncompute envy-free solutions for fractional assignment problems.\nSuppose $\\mathcal{C}$ is an algorithm that computes envy-free solutions\n$(\\bm{x}, \\bm{p})$ for any given fractional assignment problem.\nLet $\\mathcal{A}$ be an algorithm for a multi-parameter mechanism design problem\n$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$.\nWe will convert algorithm $\\mathcal{A}$ into to a mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$:\n\n\\begin{enumerate}\n\t\\item For each agent $i$ \\hspace*{\\fill} {\\bf (Pre-computation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Estimate the values \n\t\t\t\t$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \n\t\t\t\tfor the induced assignment problem of agent $i$\n\t\t\t\twith respect to $\\mathcal{A}$. Let $\\hat{\\bm{w}}_i = \n\t\t\t\t\\{\\hat{w}^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ denote the\n\t\t\t\testimated values.\n\t\t\t\\item Use $\\mathcal{C}$ to solve the induced assignment problems \n\t\t\t\twith estimated values. Let\n\t\t\t\t$(\\bm{x}_i, \\bm{p}_i)$ denote the solution by $\\mathcal{C}$ for\n\t\t\t\tthe induce assignment problem of agent $i$.\n\t\t\t\\item Let $y^t_i = f_i(t) - \\sum_{s=1}^\\ell x^{st}_i$ denote the \n\t\t\t\tunallocated supply of virtual product $t$ in solution \n\t\t\t\t$(\\bm{x}_i, \\bm{p}_i)$ for all $1 \\leq i \\leq n$ and \n\t\t\t\t$1 \\leq t \\leq \\ell$. \n\t\t\t\\item Let $y_i = \\sum_{t=1}^\\ell y^t_i$ \n\t\t\t\tdenote the total amount of unallocated virtual products \n\t\t\t\tin $(\\bm{x}_i, \\bm{p}_i)$ for all $1 \\leq i \\leq n$.\n\t\t\\end{enumerate}\n\t\\item Manipulate the valuations with intermediate algorithm $\\mathcal{B}_i$,\n\t\t$1 \\leq i \\leq n$, as follows:\n\t\t\\hspace*{\\fill} {\\bf (Decoupling)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Suppose the reported valuation of agent $i$ is $v'_i = \n\t\t\t\tv^s_i$.\n\t\t\t\\item Let $\\widetilde{v}_i = \\mathcal{B}_i(v'_i) = v^t_i$ with \n\t\t\t\tprobability $x^i_{st} \/ f_i(s)$ for $1 \\leq t \\leq \\ell$.\n\t\t\t\\item With probability $1 - \\sum_t x^{st}_i \/ f_i(s)$, the\n\t\t\t\tmanipulated valuation $\\widetilde{v}_i$ is unspecified \n\t\t\t\tin the previous step. In this\n\t\t\t\tcase, let $\\widetilde{v}_i = v^t_i$ with probability \n\t\t\t\t$y^t_i \/ y_i$ for $1 \\leq t \\leq \\ell$.\n\t\t\\end{enumerate}\n\t\\item Allocate services as follows: \\hspace*{\\fill} {\\bf (Allocation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Compute a tentative allocation\n\t\t\t\t$\\widetilde{\\bm{S}} = \n\t\t\t\t(\\widetilde{S}_1, \\dots, \\widetilde{S}_n) = \n\t\t\t\t\\mathcal{A}(\\widetilde{\\bm{v}})$.\n\t\t\t\\item For each agent $i$, let $S_i = \\widetilde{S}_i$ if the\n\t\t\t\tmanipulated valuation $\\widetilde{v}_i$ is specified in\n\t\t\t\tstep 2b). Let $S_i = \\phi$ otherwise. Allocate services\n\t\t\t\taccording to $\\bm{S}$.\n\t\t\t\\item For each agent $i$, suppose the reported valuation is \n\t\t\t\t$v'_i = v^s_i$ and the manipulated valuation is \n\t\t\t\t$\\widetilde{v}_i = v^t_i$, charge agent $i$ with price \n\t\t\t\t$p^t_i v^s_i(S_i) \/ \\hat{w}^{st}_i$.\n\t\t\\end{enumerate}\n\\end{enumerate}\n\nThe following theorem states the above meta-reduction scheme converts \nalgorithms into IR and BIC mechanisms.\n\n\\begin{theorem}\n\tSuppose the algorithm $\\mathcal{C}$ always provides envy-free solutions.\n\t\\begin{enumerate}\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$\n\t\t\tare accurate, then mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is IR and BIC.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that for\n\t\t\tany $1 \\leq s, t \\leq \\ell$, $\\hat{w}^{st}_i \\in [(1 - \\epsilon) \n\t\t\tw^{st}_i, (1 + \\epsilon) w^{st}_i]$, then $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is \n\t\t\tIR and $4 \\epsilon v_{max}$-BIC.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that\n\t\t\tfor any $1 \\leq s, t \\leq \\ell$, $\\hat{w}^{st}_i \\in \n\t\t\t[w^{st}_i - \\epsilon, w^{st}_i + \\epsilon]$, then \n\t\t\t$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is IR and $4 \\epsilon$-BIC.\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\tLet us outline the proof for part 1. Proofs of the other two parts are\n\tcalculations along the same line.\n\n\tNote that $p^t_i \\leq w^{st}_i$ for all $1 \\leq i \\leq n$ and \n\t$1 \\leq s, t \\leq \\ell$.\n\tThe mechanism is IR because for any $1 \\leq i \\leq n$ and \n\t$1 \\leq s \\leq \\ell$ the utility for an agent $i$ with valuation $v^s_i$ \n\tin any realization is\n\t$$v^s_i(S_i) - p^t_i \\frac{v^s_i(S_i)}{w^{st}_i} \\geq 0 \\enspace.$$\n\t\n\tNext, we will show that mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is BIC.\n\tWe first verify that the intermediate algorithms $\\mathcal{B}_i$,\n\t$1 \\leq i \\leq n$, satisfy the constraint \\eqref{eq:interconstraint}. \n\tFor any agent $i$, if its valuation $v_i$\n\tis drawn from distribution $F_i$, then the probability that the\n\tmanipulated valuation $\\widetilde{v}_i = \\mathcal{B}_i(v_i) = v^t_i$ is \n\t\\begin{eqnarray*}\n\t\t\\sum_{s=1}^\\ell f_i(s) \\, \\left[\\frac{x^{st}_i}{f_i(s)} + \n\t\t\\left(1 - \n\t\t\\sum_{r=1}^\\ell \\frac{x^{sr}_i}{f_i(s)} \\right) \\, \\frac{y^t_i}{y_i}\n\t\t\\right]\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\left( \\sum_{s=1}^\\ell f_i(s) - \n\t\t\\sum_{s=1}^\\ell \\sum_{r=1}^\\ell x^{sr}_i \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\left( \\sum_{r=1}^\\ell f_i(r) - \n\t\t\\sum_{r=1}^\\ell \\sum_{s=1}^\\ell x^{sr}_i \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\sum_{r=1}^\\ell \\left( f_i(r) - \n\t\t\\sum_{s=1}^\\ell x^i_{sr} \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\sum_{r=1}^\\ell y^r_i \\,\n\t\t\\frac{y^t_i}{y_i} = \\sum_{s=1}^\\ell x^{st}_i + y^t_i \n\t\t= f_i(t) \\enspace.\n\t\\end{eqnarray*}\n\n \tThus, we get that for each agent $i$, the intermediate algorithms \n\t$\\mathcal{B}_j$, $1 \\leq j \\leq n$ and $j \\neq i$, are transparent to it.\n\tSo the expected value of agent $i$ of the service it gets, when\n\tits genuine valuation is $v_i = v^s_i$ and the manipulate valuation, is \n\t$\\widetilde{v}_i = v^t_i$ is exactly \n\t$$w^{st}_i = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i)} \\enspace.$$ \n\n\tHence, the expected value of agent $i$ of the servie it gets, when its\n\tgenuine valuation is $v_i = v^s_i$ and the reported valuation is $v'_i\n\t= v^t_i$, is \n\t$$\\widetilde{w}^{st}_i = \n\t\\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, w^{sr}_i \\enspace.$$\n\t\n\tAnd the expected price for agent $i$ when the reported valuation is \n\t$v'_i = v^t_i$ is \n\t$$\\widetilde{p}^t_i = \n\t\\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\\mathcal{A}(v^r_i, \\bm{v}_{-i})}{p^r_i \\, \\frac{v^t_i(S_i)}{w^{tr}_i}} \n\t= \\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} p^r_i \\enspace.$$\n\n\tThus, the the expected utility of agent $i$, when its genuine valuation\n\tis $v_i = v^s_i$ and its reported valuation is $v'_i = v^t_i$, is \n\t$$\\widetilde{w}^{st}_i - \\widetilde{p}^t_i = \\sum_{r=1}^\\ell\n\t\\frac{x^{tr}_i}{f_i(t)} \\, (w^{sr}_i - p^r_i) \n\t\\leq \\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, \\max_k \n\t\\{w^{sk}_i - p^k_i\\} = \\max_k \\{w^{sk}_i - p^k_i\\} \\enspace.$$\n\n\tSince $\\bm{p}_i$ are chosen to be the envy-free prices, we have that\n\t$x^{sr}_i > 0$ only if $w^{sr}_i - p^r_i = \\max_k \\{w^{sk}_i - p^k_i\\}$.\n\tHence, when agent $i$ reports its valuation truthfully, that is, $s = t$,\n\tthe above inequality if tight. Moreover, the above utility is always\n\tnon-negative. So mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is BIC.\n\\end{proof}\n\nMoreover, the revenue and residual surplus of mechanism \n$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ \nis related to the social welfare and revenue of the induced assignment \nproblems as stated in following proposition.\n\n\\begin{proposition}\n\tThe expected revenue (residual surplus) of the mechanism \n\t$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ equals the sum of the revenue (residual surplus) \n\tof the manipulated assignment problems.\n\\end{proposition}\n\nBy choosing proper allocation algorithm $\\mathcal{C}$, we can obtain\ntheoretical bounds for the revenue or residual surplus in the manipulated\ninduced assignment problems and thus theoretical bounds for mechanism\n$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$.\n\n\\subsection{Assignment algorithms.}\n\nIn this section, we will introduce two algorithms for computing envy-free \nsolutions for the induced assignment problems. These two algorithms\nprovides theoretical bounds for revenue and residual surplus.\n\n\\paragraph{Revenue.} \nThe first algorithm provides revenue that is a $\\Omega(1\/\\log(1\/\\delta))$\nfraction of $SW^\\mathcal{A}$, the social welfare by algorithm $\\mathcal{A}$. The idea is\nto introduce proper reserve prices to the induced assignment problems by\nredundant virtual buyers. This is inspired by the techniques by\nGuruswami et al.~\\cite{guruswami2005profit}.\nFor the induced assignment problem of agent $i$, $1 \\leq i \\leq n$, \nthe assignment algorithm $\\mathcal{C}_R$ for revenue maximization proceeds as\nfollows:\n\\begin{enumerate}\n\t\\item Find the social welfare maximizing allocation $\\bm{x}_i = \n\t\t\\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$.\n\t\\item Suppose $u_{max}$ is the maximal valuation \n\t\tamong the virtual buyer-product pair $(s, t)$ with non-zero $x^{st}_i$,\n\t\tthat is, \n\t\t$$u_{max} = \\max \\{w^{st}_i : 1 \\leq s, t \\leq \\ell, x^{st}_i > 0\\}\n\t\t\\enspace.$$\n\t\\item Recall that $\\delta = \\min \\{f_i(t) : 1 \\leq i \\leq n, 1 \\leq t \\leq\n\t\t\\ell, f_i(t) > 0\\}$ denotes the granularity of the prior distribution. \n\t\tFor $1 \\leq k \\leq \\log(1 \/ \\delta)$:\n\t\t\\begin{enumerate}\n\t\t\t\\item Consider the following variant\n\t\t\t\tof the induced assignment instance of agent $i$: \\\\\n\t\t\t\tFor each virtual product $1 \\leq t \\leq \\ell$, add a \n\t\t\t\tdummy virtual buyer with demand $1 + \\delta$ and value \n\t\t\t\t$u_k = u_{max} \/ 2^k$ for virtual product $t$\n\t\t\t\tand value $0$ for other virtual products.\n\t\t\t\\item Find social welfare maximizing allocation $\\bm{x}_{ik}$ and \n\t\t\t\tenvy-free prices $\\bm{p}_{ik}$ for this variant.\n\t\t\t\\item Let $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ be the projection\n\t\t\t\tof $(\\bm{x}_{ik}, \\bm{p}_{ik})$ on the original induced \n\t\t\t\tassignment problem of agent $i$, that is, for any \n\t\t\t\t$1 \\leq s, t \\leq \\ell$,\n\t\t\t\t$\\hat{x}^{st}_{ik} = x^{st}_{ik}$, $\\hat{p}^t_{ik} = p^t_{ik}$.\n\t\t\\end{enumerate}\n\t\\item Return the $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, \n\t\t$1 \\leq k \\leq \\log(1\/\\delta)$, with best revenue.\n\\end{enumerate}\n\n\\begin{lemma}\n\tAssignment algorithm $\\mathcal{C}_R$ always return an envy-free solution \n\t$(\\bm{x}, \\bm{p})$. The revenue is at least a\n\t$\\Omega(1\/\\log(1\/\\delta))$ fraction of the optimal social welfare of\n\tthe assignment problem.\n\\end{lemma}\n\n\\begin{proof}\n\tThe envy-freeness follows from the fact that \n\t$(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, $1 \\leq k \\leq \\log(1\/\\delta)$,\n\tare projections of envy-free solutions and thus are also envy-free.\n\t\n\tNow we consider the revenue by $\\mathcal{C}_R$. We let $r_k$ denote\n\tthe revenue by solution $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$.\n\tNote that in $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, all prices are at \n\tleast $u_k$ and the amount of virtual products that are sold is at least\n\t$\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i$. Hence, we have \n\t$$r_k \\geq w_k \\sum_{s, t : w_{st} \\geq u_k} x^{st}_i \\enspace.$$\n\n\tNote that if we extend the definition of $u_k$ and let $u_k = u_{max} \/ \n\t2^k$ for all non-negative integer $k$, then we have\n\t\\begin{eqnarray}\n\t\t\\sum_{k=1}^\\infty u_k \\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \n\t\t\\notag \n\t\t& = & \\sum_{k=1}^\\infty (u_{k - 1} - u_k) \n\t\t\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \\notag \\\\\n\t\t& = & \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i \n\t\t\\sum_{k : w^{st}_i \\geq u_k} (u_{k-1} - u_k) \\notag \\\\\n\t\t& = & \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i \n\t\t\\max_k \\{u_{k-1} : w^{st}_i \\geq u_k\\} \\notag \\\\\n\t\t& \\geq & \\sum_{s, t} x^{st}_i w^{st}_i \\enspace. \\label{eq:rev1}\n\t\\end{eqnarray}\n\n\tOn the other hand, the contribution of the tail is small compared to\n\tthe social welfare.\n\t\\begin{equation}\n\t\t\\label{eq:rev2}\n\t\t\\sum_{k=\\log(1\/\\delta)+1}^{\\infty} u_k \n\t\t\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \n\t\t\\leq \\sum_{k = \\log(1\/\\delta) + 1}^{\\infty} w_k \n\t\t\\leq \\frac{\\delta w_{max}}{2} \n\t\t\\leq \\frac{\\sum_{s, t} x^{st}_i w^{st}_i}{2} \\enspace. \n\t\\end{equation}\n\n\tThe last inequality holds because allocating the most valuable \n\tvirtual product the one of the virtual buyer is a feasible allocation.\n\tHence, consider the difference of the above formulas, \n\t$\\eqref{eq:rev1} - \\eqref{eq:rev2}$, and we get that\n\t$$\\sum_{k = 1}^{\\log(1\/\\delta)} r_k \\geq \\sum_{k = 1}^{\\log(1\/\\delta)} u_k\n\t\\sum_{s, t : w_{st} \\geq u_k} x^{st}_i \\geq \n\t\\frac{\\sum_{s, t} x^{st}_i w^{st}_i}{2} \\enspace. $$\n\n\tThus, by pigeon-hole-principle at least one of the assignment \n\t$(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ provides \n\trevenue that is at least a $1\/2\\log(1\/\\delta)$ fraction of \n\tthe social welfare.\n\\end{proof}\n\nThe above lemma leads to the following results for revenue maximization.\n\n\\begin{proposition}\n\tSuppose the social welfare given by allocation algorithm $\\mathcal{A}$ is\n\t$SW^\\mathcal{A}$, the mechanism $\\mathcal{M}^{\\mathcal{C}_R}_\\mathcal{A}$ guarantees at least\n\t$\\Omega(SW^\\mathcal{A} \/ \\log(1\/\\delta))$ of revenue.\n\\end{proposition}\n\n\\paragraph{Complementary lower bound.}\nThe approximation ratio with respect to $SW^\\mathcal{A}$ is tight due to the \nfollowing example. Consider the auction problem with only one agent and one\nitem. Suppose with probability $1 \/ 2^k$ the agent has value $2^k$ for the item \nfor $k = 1, 2, \\dots, \\log(1 \/ \\delta)$. And with probability $\\delta$, the\nagent has value $0$ for the item. In this example, the granularity\nof the prior distribution is $\\delta$. \nThe optimal social welfare is \n$\\sum_{k=1}^{\\log(1\/\\delta)} \\frac{1}{2^k} \\, 2^k = \\log(1\/\\delta)$. \nBut no BIC mechanism can achieve revenue better than $1$.\n\n\\paragraph{Residual surplus.} \nWe turn to the residual surplus maximization problem. Note that revenue\nand residual surplus are symmetric in the induced assignment problems. \nWe will use the following assignment algorithm $\\mathcal{C}_{RS}$\nbased on the same idea we use for the revenue maximization algorithm.\n\nThe residual surplus maximizing envy-free algorithm $\\mathcal{C}_{RS}$ is as\nfollows:\n\\begin{enumerate}\n\t\\item Find the social welfare maximizing allocation $\\bm{x}_i = \n\t\t\\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$.\n\t\\item Suppose $u_{max}$ is the maximal valuation \n\t\tamong the virtual buyer-product pair $(s, t)$ with non-zero $x^{st}_i$,\n\t\tthat is, \n\t\t$$u_{max} = \\max \\{w^{st}_i : 1 \\leq s, t \\leq \\ell, x^{st}_i > 0\\}\n\t\t\\enspace.$$\n\t\\item Recall that $\\delta = \\min \\{f_i(t) : 1 \\leq i \\leq n, 1 \\leq t \\leq\n\t\t\\ell, f_i(t) > 0\\}$ denotes the granularity of the prior distribution. \n\t\tFor $1 \\leq k \\leq \\log(1 \/ \\delta)$:\n\t\t\\begin{enumerate}\n\t\t\t\\item Consider the following variant\n\t\t\t\tof the induced assignment instance of agent $i$: \\\\\n\t\t\t\tFor each virtual buyer $1 \\leq t \\leq \\ell$, add a \n\t\t\t\tdummy virtual product with demand $1 + \\delta$ and value \n\t\t\t\t$u_k = u_{max} \/ 2^k$ for virtual buyer $t$\n\t\t\t\tand value $0$ for other virtual buyer.\n\t\t\t\\item Find social welfare maximizing allocation $\\bm{x}_{ik}$ and \n\t\t\t\tenvy-free prices $\\bm{p}_{ik}$ for this variant.\n\t\t\t\\item Let $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ be the projection\n\t\t\t\tof $(\\bm{x}_{ik}, \\bm{p}_{ik})$ on the original induced \n\t\t\t\tassignment problem of agent $i$, that is, for any \n\t\t\t\t$1 \\leq s, t \\leq \\ell$,\n\t\t\t\t$$\\hat{x}^{st}_{ik} = x^{st}_{ik} \\quad , \\quad \n\t\t\t\t\\hat{p}^t_{ik} = p^t_{ik} \\enspace.$$\n\t\t\\end{enumerate}\n\t\\item Return the $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, \n\t\t$1 \\leq k \\leq \\log(1\/\\delta)$, with best revenue.\n\\end{enumerate}\n\nThe proofs of the following lemma and theorem is almost identical to the \nrevenue maximization part so we omit the proofs here.\n\n\\begin{lemma}\n\tAssignment algorithm $\\mathcal{C}_{RS}$ always return an envy-free solution \n\t$(\\bm{x}, \\bm{p})$. The residual surplus is at least a\n\t$\\Omega(1\/\\log(1\/\\delta))$ fraction of the optimal social welfare of\n\tthe assignment problem.\n\\end{lemma}\n\n\\begin{proposition}\n\tSuppose the social welfare given by allocation algorithm $\\mathcal{A}$ is\n\t$SW^\\mathcal{A}$, the mechanism $\\mathcal{M}^{\\mathcal{C}_{RS}}_\\mathcal{A}$ guarantees at least\n\t$\\Omega(SW^\\mathcal{A} \/ \\log(1\/\\delta))$ of residual surplus.\n\\end{proposition}\n\n\\section{Application in combinatorial auctions}\n\nIn this section we will briefly illustrates how to use the reduction for social \nwelfare in this paper to derive a combinatorial auction mechanism that matches \nthe best algorithmic result. \n\n\\paragraph{Combinatorial auctions.} In the combinatorial auctions, we consider\na principal with $m$ items (exactly one copy of each item) and $n$ agents. \nEach agent has a private valuation $v_i \\sim F_i$ for subsets of items. \nThe goal is to design a protocol to \nallocate the items and to charge prices to the agents.\n\n\\medskip\n\nWe will show the following corollaries of our reduction for social welfare.\n\n\\begin{corollary}\n\tFor combinatorial auctions with sub-additive (or fractionally sub-additive) \n\tagents where the prior distributions have finite and poly-size \n\tsupports, there is a $\\left(\\frac{1}{2} - \\epsilon\\right)$-approximate\n\t(or $\\left(1- \\frac{1}{e}-\\epsilon\\right)$-approximate respectively),\n\t$\\epsilon v_{max}$-BIC, and IR mechanism for social welfare maximization.\n\tThe running time is polynomial in the input size and $1 \/ \\epsilon$. \n\\end{corollary}\n\n\\paragraph{Algorithm.}\n\nWe will consider a variant of the LP-based algorithms by Feige \n\\cite{feige2006maximizing} and use the reduction for social welfare to\nconvert it into an IR and $\\epsilon v_{max}$-BIC mechanism. More concretely,\nwe will consider the Bayesian version of the standard \nsocial welfare maximization linear program (CA):\n\\begin{align*}\n\t\\text{Maximize} ~ ~ \\sum_i \\sum_t \\sum_S & ~ f_i(t) \\, v^t_i(S) \\,\n\tx_{i, t, S} & & \\text{s.t.} \\\\\n\t\\sum_i \\sum_t \\sum_{S : j \\in S} f_i(t) \\,x_{i, t, S} & ~ \\leq ~ 1 & &\n\t\\forall j \\\\\n\t\\sum_S x_{i, t, S} & ~ \\leq ~ 1 & & \\forall i, t \\\\\n\tx_{i, t, S} & ~ \\geq ~ 0 & & \\forall i, t, S\n\\end{align*}\n\nIn this LP, $x_{i, t, S}$ denote the probability that agent $i$ is allocated \nwith a subset of items $S$ conditioned on its valuation is $v^t_i$.\nThis LP can be solved in polynomial time by the standard primal dual \ntechnique via demand queries. See \\cite{dobzinski2005approximation} for more\ndetails. We let $LP^*$ denote the optimal value of this LP.\nMoreover, for any basic feasible optimal solution of the above LP,\nthere are at most $nm\\ell$ non-zero entries since there are only $nm\\ell$\nnon-trivial constraints. Hence, we have the following lemma:\n\n\\begin{lemma}\n\tIn poly-time we can find an optimal solution $\\bm{x}^*$ to (CA) with at \n\tmost $nm\\ell$ non-zero entries.\n\\end{lemma}\n\nNext, we will filter this solution $\\bm{x}^*$ by removing insignificant \nentries. We let $\\hat{x}_{i, t, S} = x^*_{i, t, S} < \\epsilon \/ n m \\ell$.\nNote that $LP^* \\geq f_i(t) v^t_i(S)$ for any $i$, $t$, and $S$ since always\nallocating subset $S$ to agent $i$ is a feasible allocation. We get that\n$\\hat{\\bm{x}}$ is a feasible solution to (CA) with value at least \n$(1 - \\epsilon) LP^*$.\n\nThen, we will use the rounding algorithms by Feige \\cite{feige2006maximizing}\nto get a $\\frac{1}{2}$-rounding\nfor sub-additive agents and a $\\left(1-\\frac{1}{e}\\right)$-rounding for\nfractionally sub-additive agents: \n\\begin{enumerate}\n\t\\item Allocate a tentative subset of items\n\t\t$\\widetilde{S}_i$ to each agent $i$, $1 \\leq i \\leq n$, according\n\t\tto the reported valuation $v'_i = v^t_i$ and \n\t\t$\\hat{x}_{i, t, \\widetilde{S}_i}$.\n\t\\item Resolve conflicts properly by choosing $S_i \\subseteq \n\t\t\\widetilde{S}_i$ so that $\\bm{S} = (S_1, \\dots, S_n)$ is a feasible\n\t\tallocation.\n\\end{enumerate}\n\nBy extending Feige's original proof, we can show that there is a randomized\nalgorithm for choosing $\\bm{S}$ such that for sub-additive agents, we have:\n\\begin{equation}\n\t\\label{eq:suba}\n\t\\e{\\bm{v}_{-i}, \\bm{S}}{v_i(S_i)} \\geq \\frac{1}{2} \\, v_i(\\widetilde{S}_i)\n\t\\enspace.\n\\end{equation}\n\nAnd for fractionally sub-additive agents, we have:\n\\begin{equation}\n\t\\label{eq:fracsuba}\n\t\\e{\\bm{v}_{-i}, \\bm{S}}{v_i(S_i)} \\geq \n\t\\left(1 - \\frac{1}{e}\\right) \\, v_i(\\widetilde{S}_i)\n\t\\enspace.\n\\end{equation}\n\nWe will omit the proof in this extended abstract.\nWe denote the above algorithm as $\\mathcal{A}$. Then, $\\mathcal{A}$ provides \n$\\left(\\frac{1}{2} - \\epsilon\\right)$-approximation for sub-additive agents \nand $\\left(1 - \\frac{1}{e} - \\epsilon\\right)$-approximation for \nfractionally sub-additive agents.\n\n\\paragraph{Estimating values.}\n\nBy Theorem \\ref{thm:welfarereduction} and \\ref{thm:otherreduction}, we only\nneed to show how to estimate the values $\\bm{w}_i$, $1 \\leq i \\leq n$, for\nthe induced assignment problem of agent $i$ efficiently. Further, by\nLemma \\ref{lemma:relative}, we can efficiently estimate the values \n$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$, $1 \\leq i \\leq n$, if\nthe following lemma holds.\n\n\\begin{lemma}\n\tFor any $1 \\leq i \\leq n$, and any $1 \\leq s, t \\leq \\ell$,\n\t$$\\frac{\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t{\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t\\leq \\sqrt{\\frac{4nm\\ell}{\\epsilon}} \\enspace.$$\n\\end{lemma}\n\n\\begin{proof}\n\tBy inequalities \\eqref{eq:suba} and \\eqref{eq:fracsuba}, we get that\n\tconditioned on $\\widetilde{S}_i$ being chosen as the tentative set,\n\t\\begin{equation*}\n\t\t{\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}}\n\t\t\\geq \\frac{1}{2} \\, v^s_i\\left(\\widetilde{S}_i\\right) \\enspace.\n\t\\end{equation*}\n\n\tWe also have that\t\n\t$$\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i) : \\widetilde{S}_i} \\leq \\max \\left\\{v^s_i(S_i) : \n\t\\widetilde{S}_i\\right\\} \\leq v^s_i(\\widetilde{S}_i) \\enspace.$$\n\n\tHence, \n\t\\begin{align*}\n\t\t\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i)}^2\n\t\t= \\, & \\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\,\n\t\t\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}^2 \\\\\n\t\t\\leq \\, & \\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\, v^s_i(\\widetilde{S}_i)^2 \\\\\n\t\t\\leq \\, & \\frac{1}{\\min \\left\\{\\hat{x}_{i, t, \\widetilde{S}_i} > 0\n\t\t\\right\\}} \\, \\left(\\sum_{i, t, \\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\, v^s_i(\\widetilde{S}_i)\\right)^2 \\\\\n\t\t\\leq \\, & \\frac{nm\\ell}{\\epsilon} \\, \\left(\\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\,\n\t\t\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}\\right)^2 \\\\\n\t\t\\leq \\, & \\frac{4nm\\ell}{\\epsilon} \\, \n\t\t\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}^2 \n\t\t\\enspace.\n\t\\end{align*}\n\\end{proof}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{introduction}Introduction}\nWhen the mass of a quark is significantly larger than the quantum chromodynamics (QCD) scale parameter, $\\Lambda_{QCD} \\sim 250$~MeV, we categorize it as a heavy quark \\cite{Behnke:2015qja}. The production of heavy quarks in photoproduction ($\\gamma p$) and deep inelastic scattering (DIS) of ${e^\\pm}p$ was one of the main tasks at HERA. The only heavy quarks kinematically accessible at HERA were beauty and charm quarks, and investigation of the impact of charm quark cross section H1-ZEUS combined data \\cite{Abramowicz:1900rp} on simultaneous determination of parton distribution functions and the strong coupling, $\\alpha_s(M^2_Z)$, is the main topic of this analysis. In deep inelastic ${e^\\pm}p$ scattering we can approximate the ratio of photon couplings corresponding to a heavy quark, $Q_h, h=b, c$, by\n\\begin{equation} \\label{eq:hfrac}\nf(h) \\sim \\frac{Q_h^2}{\\Sigma{Q_q^2}} ,\n\\end{equation}\nwhere $Q_h=\\frac{1}{3},~\\frac{2}{3}$ are the beauty and charm electric charges, respectively, and $Q_q$~, with $q=u,d,s,c,b$~, represent the kinematically accessible quark flavours. \n\n Now, for the charm quark we have\n\\begin{equation} \\label{eq:cfrac}\nf(c) \\sim \\frac{Q_c^2}{Q_d^2+Q_u^2+Q_s^2+Q_c^2+Q_b^2}\n = \\frac{4}{11} \\simeq 0.36~.\n\\end{equation}\nFrom Eq. \\ref{eq:cfrac}~ we see that up to 36 percent of the cross sections at HERA originate from processes with charm quarks in the final state. This is our main motivation to investigate the impact of only charm quarks on simultaneous determination of parton distribution functions or their uncertainties and the strong coupling, $\\alpha_s(M^2_Z)$.\n\nThe ratio $f(c) \\simeq 0.36~$ implies that charm quarks are an integral part of the quark-antiquark sea within the proton. On the other hand, the proton has no net charm flavour number, which in turn implies that the charm quarks within the proton can only arise in pairs of $c\\overline{c}$. Since the charm-quark mass is about $1.5$ GeV~, at the low-energy limit the $c\\overline{c}$ pairs are considerably heavier than that to have a contribution within the proton. \n\n Although consideration of so-called intrinsic charm (IC) \\cite{Brodsky:1980pb} may alter this simple view of the heavy flavour content of the proton, at present there is no evidence for the existence of such a contribution from HERA data. Therefore, in this analysis the charm quarks within the proton are as usual considered as virtual quarks, which in turn arise as fluctuations of probing the gluon content of the proton.\n \n The charm PDFs play an important role in hadronic collisions and cause photons to emerge from hard parton-parton interactions in association with one or more charm quark jets. Clearly, to study and analyse these processes, we need the charm PDFs, which in turn have sizeable uncertainties. A series of experimental measurements involving charm (or beauty) and photon final states have recently been published by the CDF and D0 Collaborations \\cite{Abazov:2012ea,Abazov:2009de,D0:2012gw,Abazov:2014hoa,Aaltonen:2009wc,Aaltonen:2013ama}. \n \n \n As we noted, the charm quark mass is about $1.5$ GeV~, whereas the QCD scale is about $\\Lambda_{QCD} \\sim 0.25$ GeV~, so it is reasonable to treat the charm quark mass as a hard scale in perturbative quantum chromodynamics (pQCD) and investigate the charm mass effect in pQCD. Accordingly, in this study we use the full HERA run I and II combined data \\cite{Abramowicz:2015mha} as a new measurements of inclusive deep inelastic scattering cross sections for our base data set and then we investigate, simultaneously, the impact of charm quark cross section H1-ZEUS combined data \\cite{Abramowicz:1900rp} on the central value of the PDFs and determination of the strong coupling, $\\alpha_s(M^2_Z)$. \n \n Although the charm quark mass is large compared to the QCD scale, it is small with respect to other pQCD scales, such as the transverse momentum of a quark or a jet, $p_T$, and the virtuality of the photon, $Q^2$. This smallness leads to the logarithmic correction terms, $\\sim [\\alpha_s \\ln(p^2_{T}\/m_c^2)]^n$ and $\\sim [\\alpha_s \\ln(Q^2\/m_c^2)]^n$~, corresponding to $p_T$ and $Q^2$, respectively.\nAt present, the order of magnitude and treatment of these correction terms are open questions.\n\n The outline of this paper is as follows. In Section~2, we describe the theoretical framework of our study and discuss the reduced cross sections. We introduce the data set which we use in this QCD-analysis and discuss our methodology in Section~3. In Section~4, the impact of charm quark cross section H1-ZEUS combined data on QCD fit quality is discussed. We explain the impact of charm production data on PDFs and $\\alpha_s(M^2_Z)$ in Section~5. We present our results in Section~6 and conclude with a summary in Section~7.\n\n\\section{\\label{dis}Cross sections and parton distributions}\nIn perturbative quantum chromodynamics, the deep inelastic scattering of ${e^\\pm}p$, at the centre-of-mass energies up to $\\sqrt{s} \\simeq 320\\,$GeV at HERA, plays a central role in probing the structure of the proton, as a sea of strongly interacting quarks and gluons. For neutral current (NC) interactions, the reduced cross sections can be expressed in terms of the generalized structure functions as:\n\\begin{eqnarray}\n \\sigma_{r,NC}^{{\\pm}}&=& \\frac{d^2\\sigma_{NC}^{e^{\\pm} p}}{d{x_{\\rm Bj}}dQ^2} \\frac{Q^4 x_{\\rm Bj}}{2\\pi \\alpha^2 Y_+} \\nonumber\\\\\n &=& \\tilde{F_2} \\mp \\frac{Y_-}{Y_+} x\\tilde{F_3} -\\frac{y^2}{Y_+} \\tilde{F_{\\rm L}}~,\n \\label{eq:NC}\n\\end{eqnarray} \nwhere $Y_{\\pm} = 1 \\pm (1-y)^2$ and $\\alpha$ is the fine-structure constant which is defined \nat zero momentum transfer. The generalized structure functions $\\tilde F_2$, $\\tilde F_L$ and $\\tilde F_3$ can be expressed as linear combinations of the proton structure functions $F^{\\gamma}_2, F^{\\gamma Z}_2, F^{\\gamma Z}_3, F^Z_2$ and $F^Z_3$ as follows:\n\\begin{eqnarray} \\label{strf} \n \\tilde F_2 &=& F_2 - \\kappa_Z v_e \\cdot F_2^{\\gamma Z} + \n \\kappa_Z^2 (v_e^2 + a_e^2 ) \\cdot F_2^Z~, \\nonumber \\\\ \n \\tilde F_L &=& F_{\\rm L} - \\kappa_Z v_e \\cdot F_{\\rm L}^{\\gamma Z} + \n \\kappa_Z^2 (v_e^2 + a_e^2 ) \\cdot F_{\\rm L}^Z~, \\nonumber \\\\ \n x\\tilde F_3 &=& - \\kappa_Z a_e \\cdot xF_3^{\\gamma Z} + \n \\kappa_Z^2 \\cdot 2 v_e a_e \\cdot xF_3^Z~, \n\\end{eqnarray}\nwhere $v_e$ and $a_e$ are the vector and axial-vector weak couplings of the electron to the $Z$ boson, and $\\kappa_Z(Q^2)$ is defined as\n\\begin{eqnarray}\n\\kappa_Z(Q^2) &=& \\frac{Q^2}{(Q^2+M_Z^2)(4\\sin^2 \\theta_W \\cos^2 \\theta_W)}.\n\\end{eqnarray}\nThis analysis is based on xFitter, an open source QCD framework \n\\cite{xFitter} which is an update of the former HERAFitter package \\cite{ Sapronov}. The values of the $Z$-boson mass and the electroweak mixing angle are $M_Z=91.1876$~GeV and $\\sin^2 \\theta_W=0.23127$, respectively, and electroweak effects have been treated only at leading order (LO).\n\n In the range of low values of $Q^2$, $Q^2\\ll M_Z^2$, the $Z$ boson exchange contribution may be ignored and then the reduced NC DIS cross sections can be expressed by \n\\begin{eqnarray}\n \\sigma_{r,NC}^{{\\pm}}&= & F_2 - \\frac{y^2}{Y_{+}} F_L~.\n \\label{eq:LOWNC}\n\\end{eqnarray}\n\n Similarly, the reduced charged current (CC) deep inelastic $e^{\\pm}p$ scattering cross sections may be expressed as follows:\n \n\\begin{eqnarray}\n\\sigma_{r,CC}^{\\pm} &=&\\frac{2\\pi x_{\\rm Bj}}{G^2_F} \\left[\\frac{M^2_W+Q^2}{M^2_W}\\right]^2 \\frac{d^2\\sigma_{CC}^{e^{\\pm} p}}{d{x_{\\rm Bj}}dQ^2} \\nonumber \\\\\n&= & \\frac{Y_{+}}{2} W_2^{\\pm} \\mp \\frac{Y_{-}}{2}x W_3^{\\pm} - \\frac{y^2}{2} W_L^{\\pm}~~,\n\\end{eqnarray}\nwhere $\\tilde W_2^{\\pm}$, $\\tilde W_3^{\\pm}$ and $\\tilde W_L^{\\pm}$ are another set of structure functions and $G_F$ is the Fermi constant, which is related to the weak coupling constant $g$ and electromagnetic coupling constant $e$ by:\n\\begin{eqnarray}\nG^2_F = \\frac{e^2}{4\\sqrt{2}{\\sin ^2\\theta_W}M^2_W} = \\frac{g^2}{4M_W}.\n\\end{eqnarray}\nThe values of the Fermi constant and $W$-boson mass in the xFitter QCD framework \\cite{xFitter} are: $G_F=1.16638\\times 10^{-5} $~GeV$^{-2}$\nand $M_W=80.385$~GeV.\n\n In the quark parton model (QPM), the sums and differences\nof quark and anti-quark distributions, depending on the \ncharge of the lepton beam, can be represented by $W_2^\\pm$, $xW_3^\\pm$ structure functions, respectively, and $W_{\\rm L}^\\pm = 0$~:\n\\begin{eqnarray}\n \\label{WSF}\n W_2^{+} \\approx x\\overline{U}+xD\\,,\\hspace{0.05cm} ~~~~~~~\n W_2^{-} \\approx xU+x\\overline{D}\\,,\\hspace{0.05cm} \\nonumber \\\\ \n xW_3^{+} \\approx xD-x\\overline{U}\\,,\\hspace{0.05cm} ~~~~~~~\n xW_3^{-} \\approx xU-x\\overline{D}\\,.\n\\end{eqnarray}\nAccording to Eq.~\\ref{WSF}~ we have:\n\n\\begin{eqnarray}\n\\sigma_{r,CC}^{+} \\approx x\\overline{U}+ (1-y)^2xD \\nonumber \\\\\n\\sigma_{r,CC}^{-} \\approx xU +(1-y)^2 x\\overline{D}.\n\\end{eqnarray}\nNow it is possible to determine both the valence-quark distributions, $xu_v$ and $xd_v$, and the combined sea-quark distributions, $x\\overline{U}$ and $x\\overline{D}$, by combination of NC and CC measurements.\n\n In analogy to the inclusive NC deep inelastic $e^{\\pm}p$ cross section, the reduced cross sections for charm-quark production, $\\sigma_{red}^{C\\bar{C}}$, can be expressed by\n\\begin{eqnarray}\n\t\\sigma_{red}^{C\\bar{C}} &=& \n\t \\frac{d\\sigma^{C\\bar{C}}(e^{\\pm} p)}{d{x_{\\rm Bj}} \\, dQ^2} \\cdot \\frac{x_{\\rm Bj} \\, Q^4}{2 \\pi \\alpha^2 Y_{+}} \\nonumber \\\\ \n\t &=&F_2^{C\\bar C} \\mp \\frac{Y_{-}}{Y_{+}}x F_3^{C\\bar C} - \\frac{y^2}{Y_{+}} F_L^{C\\bar C}~, \n \\label{eq:NCheavy}\n\\end{eqnarray}\t\nwhere $Y_{\\pm} = (1 \\pm (1-y)^2)$, $\\alpha$ is the electromagnetic coupling constant, and $F_2^{C\\bar{C}}$, $xF_3^{C\\bar{C}}$ and $F_L^{C\\bar{C}}$ are charm-quark contributions to the \ninclusive structure functions $F_2$, $xF_3$ and $F_L$, respectively.\n\n In the kinematic region at HERA, the $F_2^{C\\bar{C}}$ structure function makes a dominant contribution. The $xF_3^{C\\bar{C}}$ structure function contributes only from $Z^0$ exchange and $\\gamma Z^0$, which implies that for the $Q^{2} \\ll M_{Z}^{2}$ region, this contribution may be ignored. Finally, the contribution of longitudinal charm-quark structure function, $F_L^{C\\bar{C}}$, is suppressed only for the $y^2 \\ll 1$ region which can be a few percent in the kinematic region accessible at HERA and therefore cannot be ignored.\n \n Therefore, neglecting the $xF_3^{C\\bar C}$ structure function contribution, the reduced charm-quark cross section, $\\sigma_{red}^{ C\\bar C}$, for both positron and electron beams, can be expressed by \n\\begin{eqnarray}\n \\sigma_{red}^{ C\\bar C}&=& \\frac{d^2\\sigma^{C\\bar C}(e^{\\pm}p)}{dxdQ^2} \\frac{xQ^4}{2\\pi\\alpha^2 Y_{+}} \\nonumber\\\\\n &= & F_2^{C\\bar C} - \\frac{y^2}{Y_{+}} F_L^{C\\bar C}~. \n \\label{eq:NCheavylast}\n\\end{eqnarray}\nAccordingly, at high $y$, the reduced charm-quark cross section, $\\sigma_{red}^{ C\\bar C}$, and the $F_2^{C\\bar C}$ structure function only differ by a small $F_L^{C\\bar C}$ contribution \\cite{Daum:1996ec}.\n\n In the QPM, the structure functions depend only on the $Q^2$ \nvariable and then they can be directly related to the PDFs. In QCD, however, and especially when heavy flavour production is included, the structure functions depend on both $x$ and $Q^2$ variables, \\cite{Sjostrand:2001yu,Aktas:2006hy,DeRoeck:2011na,Ball:2010de,Aaron:2011gp,Tung:2006tb,Aaron:2009aa,Engelen:1998rf,CooperSarkar:2012tx,Frixione:1993yw,Marchesini:1991ch,Jung:1993gf,Sjostrand:1985ys,Sjostrand:1986hx,Lonnblad:1992tz,Kuraev:1976ge,Ciafaloni:1987ur,Jung:2000hk,Beneke:1994rs,Agashe:2014kda,Schmidt:2012az,Alekhin:2010sv,Gao:2013wwa,Martin:1998sq,Pumplin:2002vw,Chekanov:2002pv}. In Section \\ref{methodology}, based on our methodology, we extract the PDFs as functions of $x$ and $Q^2$ variables, using full HERA run I and II combined data, with and without the charm cross section H1-ZEUS combined measurements data set included.\n\n\\section{\\label{methodology}Data Set and Methodology} \nIn this paper, we use two different data sets: the full HERA run I and II combined NC and CC DIS $e^{\\pm}p$ scattering cross sections \\cite{Abramowicz:2015mha}, and the charm production reduced cross section measurements data \\cite{Abramowicz:1900rp}. In our analysis, we choose the full HERA run I and II combined data as our base data set, and then we investigate the impact of charm production reduced cross section data on simultaneous determination of PDFs and the strong coupling, $\\alpha_s(M^2_Z)$~ in the Thorne-Roberts (RT) and Thorne-Roberts optimal (RTOPT) schemes. The kinematic ranges for these two data sets have been reported in Ref.~\\cite{Vafaee:2017nze}.\n\n We use xFitter \\cite{xFitter}, version 1.2.2, as our QCD fit framework. Using the QCDNUM package \\cite{Botje:2010ay}, version 17-01\/12, we evolved the parton distribution functions and $\\alpha_s(M^2_Z)$. In the evolution of PDFs and $\\alpha_s(M^2_Z)$, we set our theory type based on the DGLAP collinear evolution equations \\cite{DGLAP} and make several fits at leading order and next-to-leading order in the RT and RTOPT schemes.\n \n The RT scheme is a General Mass-Variable Flavour Number Scheme (GM-VFNS). Really, the RT scheme was designed to provide a smooth transition from the massive FFN scheme at low scales $Q^2 \\sim m_h^2$ to the massless ZM-VFNS scheme at high scales $Q^2 \\gg m_h^2$. However, the connection is not unique. A GM-VFNS can be defined by demanding equivalence of the $n_f = n$ (FFNS) and $n_f = n + 1$ flavour (ZM-VFNS) descriptions above the transition\npoint for the new parton distributions. Of course they are by definition identical below this point, at all orders. The RT scheme has two different variants: RT standard and RT optimal, with a smoother transition across the heavy flavour threshold region. A review of the two different schemes has been given in Ref.~\\cite{Vafaee:2017nze}.\n \n To investigate the impact of charm production reduced cross section data, we need to use the heavy flavour scheme in our analysis. Different theoretical groups use various heavy flavour schemes. For example, some theory groups such as CT10 \\cite{Lai:2010vv}, ABKM09 \\cite{Alekhin:2009vn}, and NNPDF2.1 \\cite{Ball:2008by,Mironov:2009uv} used \nS-ACOT \\cite{Collins:1998rz}, FFNS \\cite{Martin:2006qz} and FONLL \\cite{Forte:2010ta}, respectively and some other groups such as MSTW08 \\cite{Martin:2009iq} and HERAPDF1.5\/2.0 \\cite{Aaron:2009aa} used the RT (also referred to as TR) standard and optimal heavy flavour schemes \\cite{Thorne:2006qt,Thorne:2012az} to calculate the reduced charm cross sections in DIS. On the other hand, to include heavy flavour contributions, the perturbative QCD scales $\\mu_f^2$ and $\\mu_r^2$ play a central rule. Some groups such as CT10 \\cite{Lai:2010vv} and ABKM09 \\cite{Alekhin:2009vn} choose $\\mu_f^2 = \\mu_r^2=Q^2+m_C^2$ and $\\mu_f^2 = \\mu_r^2=Q^2+4m_C^2$ respectively, where $m_C$ denotes the pole mass of the charm quark, whereas the NNPDF2.1 \\cite{Ball:2008by,Mironov:2009uv}, HERAPDF1.5 \\cite{Aaron:2009aa} and MSTW08 \\cite{Martin:2009iq} groups use $\\mu_f = \\mu_r=Q$ in their heavy quark QCD approach.\n \n To include the heavy-flavor contributions, we use both RT and RTOPT schemes, and choose $\\mu_f = \\mu_r=Q$~ as the perturbative quantum chromodynamics scale for the pole mass of the charm quark $m_c=1.5 \\pm 0.15$ GeV.\n \n The last step in our QCD analysis is the minimization procedure. In this regard, we use the standard MINUIT minimization package~\\cite{James:1975dr}, as a powerful program for minimization, correlations and parameter errors. \n \n In order to minimize the influence of higher twist contributions we use kinematic cuts. In the various DIS analyses, different kinds of kinematic cuts should be applied. In this analysis we imposed a kinematic cut $Q^2$=3.5 GeV$^2$ to omit all data with $Q^2$ less than this value. The cuts on the kinematic coverage of the DIS data have been made for values of $Q^2$ between $Q^2=0.045$\\,GeV$^2$ and $Q^2=50000$\\,GeV$^2$ and values of $x_{\\rm Bj}$ \nbetween $x_{\\rm Bj}=6\\times10^{-7}$ and $x_{\\rm Bj}=0.65$. The cuts on $Q^2$ not only significantly increase the number of data points available to constrain PDFs, but also allow access to a greater range of kinematics, which in turn lead to reduced PDF uncertainties, especially at higher values of $x$.\n\n In this analysis, based on the HERAPDF approach \\cite{Abramowicz:2015mha}, we generically parameterized the PDFs of the proton, $xf(x)$, at the initial scale of the QCD evolution $Q^2_0= 1.9$ GeV$^2$ as\n\\begin{equation}\n xf(x) = A x^{B} (1-x)^{C} (1 + D x + E x^2)~~,\n\\label{eqn:pdf}\n\\end{equation}\nwhere in the infinite momentum frame, $x$ is the fraction of the proton's momentum taken by the struck parton.\n \nTo determine the normalization constants $A$ for the valence and gluon distributions, we use the QCD number and momentum sum rules. Using this functional form, Eq. \\ref{eqn:pdf} leads to the following independent combinations of parton distribution functions:\n\\begin{eqnarray}\n\\label{eq:xgpar}\nxg(x) &= & A_g x^{B_g} (1-x)^{C_g} - A_g' x^{B_g'} (1-x)^{C_g'} , \\\\\n\\label{eq:xuvpar}\nxu_v(x) &= & A_{u_v} x^{B_{u_v}} (1-x)^{C_{u_v}}\\left(1+E_{u_v}x^2 \\right) , \\\\\n\\label{eq:xdvpar}\nxd_v(x) &= & A_{d_v} x^{B_{d_v}} (1-x)^{C_{d_v}} , \\\\\n\\label{eq:xubarpar}\nx\\bar{U}(x) &= & A_{\\bar{U}} x^{B_{\\bar{U}}} (1-x)^{C_{\\bar{U}}}\\left(1+D_{\\bar{U}}x\\right) , \\\\\n\\label{eq:xdbarpar}\nx\\bar{D}(x) &= & A_{\\bar{D}} x^{B_{\\bar{D}}} (1-x)^{C_{\\bar{D}}} .\n\\end{eqnarray}\nwhere $xg(x)$ is the gluon distribution, $xu_{{v}}(x)$ and $xd_{{v}}(x)$ are the valence-quark distributions, and $x\\bar{U}(x)$ and $x\\bar{D}(x)$ are the $u$-type and $d$-type anti-quark distributions, which are identical to the sea-quark distributions. A review of HERAPDF functional form, including some more details, can be found in Ref.~\\cite{Vafaee:2017nze}.\n\n\\section{\\label{qcdfitquality}Impact of Charm Production Data on the QCD fit quality }\nWe now investigate the impact of the charm cross section H1-ZEUS combined measurements on simultaneous determination of PDFs and $\\alpha_s(M^2_Z)$. We also explain how adding these data improve the uncertainty of PDFs, reducing the error bars of some parton distributions, especially gluon distributions and some of their ratios, when the HERA run I and II combined inclusive DIS $e^{\\pm}p$ scattering cross sections data are chosen as a ``BASE''.\nTo investigate the fit quality, we use the $\\chi^2$ definition as reported in Ref.~\\cite{Vafaee:2017nze}.\n\n\n For HERA run I and II combined inclusive DIS $e^{\\pm}p$ scattering cross sections and the charm cross section H1-ZEUS combined measurements, the number of data points are 1307 and 42, respectively. Accordingly, the total number of data points for BASE and BASE plus charm, which we refer to sometimes as ``TOTAL'', are 1307 and 1349, respectively. In various configurations, the $Q^2 \\ge 1.5$\\,GeV$^2$ range was covered by the HERA run I and II combined data \\cite{Abramowicz:2015mha}. The MINUIT parameters are sensitive to the $Q^2_{min}$ value, so to get a convergent fit result we set $Q^2_{min}=3.5$ GeV$^2$, as suggested in~Ref \\cite{Abramowicz:2015mha}. Clearly, this cut on $Q^2$ omits all data with $Q^2$ less than $Q^2_{min}=3.5$~GeV$^2$ and therefore, reduces the total number of data points from 1307 and 1349 to 1145 and 1192 for the BASE and TOTAL data sets, respectively. Now, based on Table~\\ref{tab:data}, we can present our QCD fit quality as follows:\n\nfor the RT scheme:\n\\begin{eqnarray}\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\frac{1335}{1131}=1.180~~$for BASE~,} \\label{lobase}\\\\\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\bf \\frac{1389}{1178}=1.179 ~~$for TOTAL~,} \\label{lototal}\n\\end{eqnarray} \nand for the RTOPT scheme:\n\\begin{eqnarray}\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\frac{1331}{1131}=1.176~~$for BASE~,} \\label{nlobase}\\\\\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\bf \\frac{1378}{1178}=1.169 ~~$for TOTAL~,} \\label{nlototal}\n\\end{eqnarray}\nwhere dof refers to the $\\chi^2$ per degrees of freedom and is defined as the number of data points minus the number of free parameters.\nAs we can see from Eqs.~(19-22), we obtain four different values of $\\chi^2_{\\rm TOTAL}$\/dof, corresponding to four different fits, which in turn imply four different fit-qualities in some PDFs. Now, according to the relative change of $\\chi^2$, which is defined by $\\frac{\\chi^2_{\\rm RT}-\\chi^2_{\\rm RT\\;OPT}}{\\chi^2_{\\rm RT}}$ and the numerical results of Eqs.~(19-22), we see that in going from the RT scheme to the RT OPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively. Clearly these differences in fit quality imply a significant reduction of some PDF uncertainties, especially for gluon distributions, as we will explain in the next section.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n{ \\bf Order} & \\multicolumn{4}{c|}{ {\\bf NLO} } \\\\ \\hline\n { \\bf Experiment} & {$~~~~$RT BASE$~~~~$} & {RT TOTAL} & { $~~~$RTOPT BASE$~~~$} & { RTOPT TOTAL} \\\\ \\hline\n HERA I+II CC $e^{+}p$ \\cite{Abramowicz:2015mha} & 44 \/ 39& 45 \/ 39& 44 \/ 39& 44 \/ 39 \\\\ \n HERA I+II CC $e^{-}p$ \\cite{Abramowicz:2015mha} & 49 \/ 42& 49 \/ 42& 50 \/ 42& 49 \/ 42 \\\\ \n HERA I+II NC $e^{-}p$ \\cite{Abramowicz:2015mha} & 221 \/ 159& 221 \/ 159& 221 \/ 159& 221 \/ 159 \\\\ \n HERA I+II NC $e^{+}p$ 460 \\cite{Abramowicz:2015mha} & 208 \/ 204& 209 \/ 204& 210 \/ 204& 210 \/ 204 \\\\ \n HERA I+II NC $e^{+}p$ 575 \\cite{Abramowicz:2015mha} & 213 \/ 254& 213 \/ 254& 212 \/ 254& 212 \/ 254 \\\\\n HERA I+II NC $e^{+}p$ 820 \\cite{Abramowicz:2015mha} & 66 \/ 70& 66 \/ 70& 65 \/ 70& 66 \/ 70 \\\\ \n HERA I+II NC $e^{+}p$ 920 \\cite{Abramowicz:2015mha} & 422 \/ 377& 424 \/ 377& 418 \/ 377& 419 \/ 377 \\\\ \\hline\n {Charm H1-ZEUS} \\cite{Abramowicz:1900rp} & - & 40 \/ 47& - & 38 \/ 47 \\\\ \\hline\n { Correlated ${\\bf \\chi^2}$} & 111& 122& 111& 119 \\\\\\hline\n{\\bf {Total $\\bf{\\chi^2}$ \/ dof}} & ${\\bf \\frac{1335}{1131}}$ & ${\\bf \\frac{1389}{1178}}$ & ${\\bf \\frac{1331}{1131}}$ & ${\\bf \\frac{1378}{1178}}$ \\\\ \\hline\n\\hline\n \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:data}{Data sets used in our NLO QCD analysis, with corresponding partial $\\chi^2$ per data point for each data set, including $\\chi^2$ per degrees of freedom (dof) for the RT and RT OPT schemes.}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\\section{\\label{impapdfalphs}Impact of Charm Production Data on PDFs and $\\alpha_s(M^2_Z)$ }\nNow, we present the impact of charm cross section H1-ZEUS combined measurements data on simultaneous determination of PDFs and $\\alpha_s(M^2_Z)$ in the RT and RTOPT schemes and for two separate scenarios. In the first scenario we fix $\\alpha_s(M^2_Z)$ to 0.117 and develop our QCD fit analysis based on only 14 unknown free parameters, according to Eqs.~(14-18). Although in this scenario the value of $\\chi^2_{TOTAL}$ \/ dof is reduced, according to Eqs.~(19-22), from 1.180 to 1.169, we find nothing to show the impact of charm cross section H1-ZEUS combined measurements data on the PDFs. In the second scenario we consider the strong coupling $\\alpha_s(M^2_Z)$ as an extra free parameter and refit our analysis, but this time with 15 unknown free parameters. Based on the second scenario, not only do we obtain $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively, the same as the first scenario, but we also clearly find the impact of charm on the PDFs, especially on the gluon distribution. Some more details about the central role of the strong coupling in pQCD have been reported in Ref.~\\cite{Vafaee:2017nze}.\n\n In Tables \\ref{tab:pa1} and \\ref{tab:pa2}, we present next-to-leading order numerical values of parameters and their uncertainties for the \n$xu_v$, $xd_v$, sea and gluon PDFs at the input scale of $Q^2_0 = 1.9$~GeV$^2$ for the two different scenarios.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{5}{|c|}{ {\\bf First Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$, is Fixed} } \\\\ \\hline\n { \\bf Parameter} & {$~~~~$RT BASE$~~~~$} & { $~~~$RT TOTAL$~~~$} & {RTOPT BASE} & {RTOPT TOTAL} \\\\ \\hline\n ${B_{u_v}}$ & $0.723 \\pm 0.046$& $0.723 \\pm 0.044$& $0.730 \\pm 0.042$& $0.726 \\pm 0.039$ \\\\ \n ${C_{u_v}}$ & $4.841 \\pm 0.088$& $4.833 \\pm 0.087$& $4.827 \\pm 0.086$& $4.823 \\pm 0.084$ \\\\ \n $E_{u_v}$ & $13.6 \\pm 2.6$& $13.5 \\pm 2.5$& $13.1 \\pm 2.3$& $13.2 \\pm 2.1$ \\\\ \\hline\n ${B_{d_v}}$ & $0.818 \\pm 0.095$& $0.826 \\pm 0.094$& $0.825 \\pm 0.094$& $0.822 \\pm 0.095$ \\\\ \n $C_{d_v}$ & $4.16 \\pm 0.40$& $4.18 \\pm 0.39$& $4.21 \\pm 0.39$& $4.19 \\pm 0.38$ \\\\ \\hline\n $C_{\\bar{U}}$ & $8.91 \\pm 0.81$& $8.72 \\pm 0.78$& $8.90 \\pm 0.81$& $8.70 \\pm 0.80$ \\\\ \n $D_{\\bar{U}}$ & $17.7 \\pm 3.3$& $16.4 \\pm 3.0$& $17.6 \\pm 3.3$& $16.5 \\pm 3.2$ \\\\ \n $A_{\\bar{D}}$ & $0.158 \\pm 0.011$& $0.160 \\pm 0.011$& $0.1561 \\pm 0.0098$& $0.1594 \\pm 0.0099$ \\\\ \n $B_{\\bar{D}}$ & $-0.1682 \\pm 0.0082$& $-0.1666 \\pm 0.0080$& $-0.1760 \\pm 0.0074$& $-0.1732 \\pm 0.0073$ \\\\ \n $C_{\\bar{D}}$ & $4.2 \\pm 1.3$& $4.5 \\pm 1.3$& $4.0 \\pm 1.2$& $4.3 \\pm 1.3$ \\\\\\hline\n $B_g$ & $-0.11 \\pm 0.16$& $-0.12 \\pm 0.15$& $-0.07 \\pm 0.13$& $-0.09 \\pm 0.12$ \\\\ \n $C_g$ & $11.2 \\pm 1.7$& $10.7 \\pm 1.4$& $12.3 \\pm 1.7$& $11.8 \\pm 1.5$ \\\\ \n $A_g'$ & $2.1 \\pm 1.5$& $1.9 \\pm 1.2$& $2.9 \\pm 1.6$& $2.7 \\pm 1.1$ \\\\ \n ${B_g'}$ & $-0.206 \\pm 0.079$& $-0.216 \\pm 0.075$& $-0.142 \\pm 0.083$& $-0.162 \\pm 0.098$ \\\\ \n \\hline \n {${ \\alpha_s(M^2_Z)}$} & $ 0.1176$ & $ 0.1176$ & $ 0.1176$ & $0.1176$ \\\\ \\hline\n\\hline\n \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:pa1}{ {The NLO numerical values of parameters and their uncertainties for the $xu_v$, $xd_v$, $x\\bar u$, $x\\bar d$, $x\\bar s$ and $xg$ PDFs at the initial scale of $Q^2_0 = 1.9$~GeV$^2$ in the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed to $0.117$.}}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{5}{|c|}{ {\\bf Second Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$, is Free} } \\\\ \\hline\n { \\bf Parameter} & {$~~~~$RT BASE$~~~~$} & { $~~~$RT TOTAL$~~~$} & {RTOPT BASE} & {RTOPT TOTAL} \\\\ \\hline\n ${B_{u_v}}$ & $0.712 \\pm 0.047$& $0.725 \\pm 0.048$& $0.710 \\pm 0.045$& $0.709 \\pm 0.043$ \\\\ \n ${C_{u_v}}$ & $4.88 \\pm 0.12$& $4.83 \\pm 0.12$& $4.89 \\pm 0.11$& $4.87 \\pm 0.10$ \\\\ \n $E_{u_v}$ & $13.9 \\pm 2.3$& $13.4 \\pm 2.2$& $13.7 \\pm 2.2$& $13.7 \\pm 2.3$ \\\\ \\hline\n ${B_{d_v}}$ & $0.811 \\pm 0.094$& $0.826 \\pm 0.096$& $0.812 \\pm 0.093$& $0.813 \\pm 0.091$ \\\\ \n $C_{d_v}$ & $4.18 \\pm 0.38$& $4.17 \\pm 0.39$& $4.24 \\pm 0.38$& $4.22 \\pm 0.39$ \\\\ \\hline\n $C_{\\bar{U}}$ & $9.09 \\pm 0.89$& $8.70 \\pm 0.90$& $9.21 \\pm 0.87$& $8.97 \\pm 0.84$ \\\\ \n $D_{\\bar{U}}$ & $18.5 \\pm 3.9$& $16.3 \\pm 3.7$& $19.2 \\pm 3.9$& $17.8 \\pm 3.5$ \\\\ \n $A_{\\bar{D}}$ & $0.160 \\pm 0.012$& $0.160 \\pm 0.011$& $0.158 \\pm 0.010$& $0.1610 \\pm 0.0100$ \\\\ \n $B_{\\bar{D}}$ & $-0.1657 \\pm 0.0099$& $-0.167 \\pm 0.010$& $-0.1728 \\pm 0.0083$& $-0.1704 \\pm 0.0079$ \\\\ \n $C_{\\bar{D}}$ & $4.4 \\pm 1.4$& $4.5 \\pm 1.4$& $4.4 \\pm 1.3$& $4.6 \\pm 1.3$ \\\\\\hline\n $B_g$ & $-0.13 \\pm 0.11$& $-0.12 \\pm 0.12$& $-0.10 \\pm 0.12$& $-0.12 \\pm 0.10$ \\\\ \n $C_g$ & $11.8 \\pm 2.2$& $10.6 \\pm 2.0$& $13.5 \\pm 2.3$& $12.7 \\pm 2.3$ \\\\ \n $A_g'$ & $2.3 \\pm 1.1$& $1.87 \\pm 0.84$& $3.4 \\pm 1.6$& $3.0 \\pm 1.6$ \\\\ \n ${B_g'}$ & $-0.217 \\pm 0.074$& $-0.215 \\pm 0.073$& $-0.164 \\pm 0.096$& $-0.182 \\pm 0.074$ \\\\ \n \\hline \n {${ \\alpha_s(M^2_Z)}$} & $0.1161 \\pm 0.0037$& $0.1178 \\pm 0.0038$& $0.1151 \\pm 0.0032$& $0.1154 \\pm 0.0028$ \\\\ \\hline\n\\hline\n \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:pa2}{ {The NLO numerical values of parameters and their uncertainties for the \n$xu_v$, $xd_v$, $x\\bar u$, $x\\bar d$, $x\\bar s$ and $xg$ PDFs at the initial scale of $Q^2_0 = 1.9$~GeV$^2$ in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is taken as an extra free parameter.}}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\n According to the numerical results in Table~\\ref{tab:pa2}, when we add the charm cross section H1-ZEUS combined measurements data to the HERA run I and II combined data, the numerical value of $\\alpha_s(M^2_Z)$ changes from $0.1161 \\pm 0.0037$ to $0.1178 \\pm 0.0038$ and from $0.1151 \\pm 0.0032$ to $0.1154 \\pm 0.0028$, for the RT and RTOPT schemes, without and with charm flavour data included, respectively. If we compare our results for $\\alpha_s(M^2_Z)$ for RT TOTAL and RTOPT TOTAL with the world average value, $\\alpha_s(M^2_Z)=0.1185 \\pm 0.0006$, which was recently reported by the PDG \\cite{Agashe:2014kda}, we find a good agreement with the world average value. Of course, it should be noted, since the PDG value of $\\alpha_s(M^2_Z)$ is extracted by global fits to a variety of experimental data, it has a much smaller uncertainty. In other words, although our QCD analysis has been performed based on only two data sets, our numerical results for the strong coupling are in good agreement with the world average value. Also, these values of strong coupling show the impact of the RT and RTOPT schemes on the determination of $\\alpha_s(M^2_Z)$, when considered as an extra free parameter. \n\n\n\\clearpage\n\\section{\\label{Results}Results}\nAccording to Table~\\ref{tab:f1}, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contributions included, respectively. Also, according to Table~\\ref{tab:f2}, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.9$~\\% and $\\sim 2.0$~\\%~improvement in the $\\alpha_s(M^2_Z)$ value, without and with the charm flavour contributions respectively. \n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{3}{|c|}{ {\\bf First Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$ is Fixed} } \\\\ \\hline\n {Scheme} & $\\chi^2_{\\rm TOTAL}\/dof$ & ${\\alpha_s(M^2_Z)}$ \\\\ \\hline \n {RT BASE} & {${ 1335\/1131}$}& $0.1176$ \\\\\n {RT TOTAL} & {${1389\/1178}$}& $0.1176$ \\\\\n {RTOPT BASE} & {${1331\/1131}$}& $0.1176$ \\\\ \n {RTOPT TOTAL} & {${1378\/1178}$}& $0.1176$ \\\\ \\hline\n\\hline\n \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:f1}{ Comparison of the numerical values of $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ for the RT and RTOPT schemes in the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed to $0.117$. RTOPT TOTAL has the best fit quality, as an impact of adding charm cross section H1-ZEUS combined measurements data to HERA I and II combined data. }}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table} \n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{3}{|c|}{ {\\bf Second Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$ is Free} } \\\\ \\hline\n {Scheme} & $\\chi^2_{\\rm TOTAL}\/dof$ & ${\\alpha_s(M^2_Z)}$ \\\\ \\hline \n {RT BASE} & {${1335\/1130}$}& $0.1161 \\pm 0.0037$ \\\\\n {RT TOTAL} & {${1389\/1177}$}& $0.1178 \\pm 0.0038$ \\\\\n {RTOPT BASE} & {${1330\/1130}$}& $0.1151 \\pm 0.0032$ \\\\ \n {RTOPT TOTAL} & {${1377\/1177}$}& $0.1154 \\pm 0.0028$ \\\\ \\hline\n\\hline\n \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:f2}{Comparison of the numerical values of $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ and ${\\alpha_s(M^2_Z)}$ for the RT and RTOPT schemes in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is taken as an extra free parameter. RTOPT TOTAL has the best fit quality and improvement in oupling, $\\alpha_s(M^2_Z)$, as an impact of adding charm cross section H1-ZEUS combined measurements data to HERA I and II combined data.}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table} \n\n\n In Fig.~\\ref{fig:1}, we illustrate the consistency of HERA measurements of the reduced deep inelastic $e^{\\pm}p$ scattering cross sections data \\cite{Abramowicz:2015mha} and charm production reduced cross section measurements data \\cite{Abramowicz:1900rp} with the theory predictions as a function of $x$ and for different values of $Q^2$. According to our QCD analysis, we have good agreement between the theoretical and experimental data. The uncertainties on the cross sections in Fig.~\\ref{fig:1} are obtained using Hessian error propagation. The corresponding $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ values for each of the data sets in Fig.~\\ref{fig:1} are listed in Table~\\ref{tab:data}. \n\n\\begin{figure*}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f1}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f2}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f3}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f4}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f5}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f6}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f7}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f8}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f9}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f10}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f11}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f12}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f13}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f14}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f15}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f16}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f17}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f18}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f19}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f20}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f21}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f22}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f23}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f24}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f25}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f26}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f27}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f28}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f29}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f30}\n\n\\caption{Illustrations of the consistency of HERA combined measurements of the reduced DIS $e^{\\pm}p$ data \\cite{Abramowicz:2015mha} and the theory predictions as a function of $x$ and for different values of $Q^2$.}\n\\label{fig:1}\n\\end{figure*}\n\n\n\n The impact of charm cross section H1-ZEUS combined measurements data on HERA I and II combined data for gluon distribution functions are shown in Figs. \\ref{fig:2} and \\ref{fig:3}, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, in the RT and RTOPT schemes and for two separate scenarios. Clearly, in the first scenario, where the strong coupling $\\alpha_s(M_Z^2)$ is fixed, we find no impact from adding charm H1-ZEUS combined data to the HERA I and II combined data. In the second scenario, however, where we consider the strong coupling $\\alpha_s(M_Z^2)$ as an extra free parameter, we clearly find the impact of adding charm H1-ZEUS combined data to the HERA I and II combined data.\n \n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g5}\n\n\\caption{The gluon PDFs as extracted for the RT scheme in two separate scenarios. These distributions are plotted at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, as a function of $x$. The upper four diagrams correspond to the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed and we find no impact of adding charm H1-ZEUS combined data to the HERA I and II combined data. The lower four diagrams correspond to the second scenario, where we consider the strong coupling, $\\alpha_s(M_Z^2)$, as an extra free parameter, clearly revealing the impact of adding charm H1-ZEUS combined data to the HERA I and II combined data.}\n\\label{fig:2}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g8}\n\n\n\\caption{The gluon PDFs as extracted for the RTOPT scheme in two separate scenarios, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, as a function of $x$. The impact of adding charm data can be seen only in the four lower diagrams, where the strong coupling, $\\alpha_s(M_Z^2)$, is considered as an extra free parameter.}\n\\label{fig:3}\n\\end{figure*}\n\n\n The partial gluon distribution functions are shown in Figs. \\ref{fig:4} and \\ref{fig:5}, at $Q^2$ = 1.9, 3, 5 and 10~GeV$^2$ in the RT and RTOPT schemes and for two separate scenarios. The impact of adding charm H1-ZEUS combined data to the HERA I and II combined data can be seen only in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is considered as an extra free parameter. \n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr4}\n\n\\caption{The partial gluon PDFs as extracted for the RT scheme in two separate scenarios. These PDFs are plotted at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 5 and 10~GeV$^2$, as a function of $x$. The upper four diagrams are based on a fixed strong coupling, and do not show the impact of adding charm flavour. The lower four diagrams, based on the second scenario, where $\\alpha_s(M_Z^2)$ is considered as an extra free parameter, clearly show this impact. }\n\\label{fig:4}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr8}\n\n\n\\caption{The partial of gluon PDFs as extracted for the RTOPT scheme, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 5 and 10~GeV$^2$, as a function of $x$. The impact of adding charm cross section H1-ZEUS combined data can bee seen in the four lower diagrams, which have been plotted based on the second scenario.}\n\\label{fig:5}\n\\end{figure*}\n\nThe total sea quark $\\Sigma$-PDFs are defined by $\\Sigma=2x(\\bar u+\\bar d+\\bar s+\\bar c)$. In Figs.~\\ref{fig:6} and \\ref{fig:7} \nwe plot the partial ratio of gluon distributions over the $\\Sigma$-PDF to show the impact of adding charm cross section H1-ZEUS combined data to HERA I and II combined data, at $Q^2$ = 4, 5, 100 and 10000~GeV$^2$ in the RT and RTOPT schemes and for the two different scenarios. Clearly, these impacts can be seen only in the second scenario.\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go4}\n\n\\caption{The partial ratio of gluon distributions over $\\Sigma$-PDFs, as extracted for the RT scheme in two separate scenarios, at $Q^2$ = 4, 5, 100 and 1000~GeV$^2$, as a function of $x$. Only the four lower diagrams, corresponding to second scenario, show the impact of charm flavour on the PDFs.}\n\\label{fig:6}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go8}\n\n\n\\caption{The partial ratio of gluon distributions over $\\Sigma$-PDFs, as extracted for the RTOPT scheme. The upper four diagrams correspond to the first scenario, while the lower four diagrams correspond to the second scenario and clearly show the impact of charm flavour data on the PDFs. }\n\\label{fig:7}\n\\end{figure*}\n\n\\clearpage\n\\section{\\label{Summary}Summary}\nUp to 36 percent of the cross sections at HERA originate from processes with charm quarks in the final state. In this QCD analysis we investigated the simultaneous impact of charm quark cross section H1-ZEUS combined data on the PDFs and on the determination of the strong coupling.\n\nWe chose the full HERA run I and II DIS charged and neutral current data as a base data set and developed our QCD analysis at next-to-leading order in both RT and RTOPT schemes and for two separate scenarios using the HERAPDF parametrization form.\n\nThe sensitivity of PDF uncertainties to reduced charm quark cross section H1-ZEUS combined data at next-to-leading order, especially when in our second scenario we take the strong coupling, $\\alpha_s(M^2_Z)$, as an extra free parameter, is reported in this QCD analysis.\n\n This analysis shows a dramatic reduction of some PDF uncertainties and good agreement of the strong coupling constant, $\\alpha_s(M^2_Z)$, with the world average value, when the reduced charm quark cross section H1-ZEUS combined data are included.\n \n As we mentioned, the strong coupling, $\\alpha_s(M^2_Z)$, plays a central role in the pQCD factorization theorem and the result of this QCD-analysis emphasis on its dramatic correlation with the PDFs reveals the impact of the charm flavour contribution.\n \n According our QCD analysis, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively.\n Also, we show that in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.9$~\\% and $\\sim 2.0$~\\%~improvement in the strong coupling value, without and with the charm flavour contribution, respectively.\n\n A standard LHAPDF library file of this QCD analysis at next-to-leading order is available and can be obtained from the author via e-mail.\n \n \n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}