Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations

摘要变化

We study two fairness objectives for allocating $m$ indivisible items among $n$ agents with subadditive valuations: the max-min objective, also known asallocation (the Santa Claus problem, which maximizes the smallest value of an agent,) and the weighted Nash social welfare, whichfor maximizes$n$ theagents weightedwith geometricsubadditive meanvaluations ofover the$m$ agents'indivisible valuesitems. For the max-minUsing allocationdemand problemqueries, we give a randomized polynomial-time approximation algorithms with factors $O(\sqrt n)$ and $O(Γ(w))$, respectively, where $Γ(w)=\exp((\ln n-approximationH(w))/2)$ algorithmand $H(w)=-\sum_i w_i\ln w_i$ for subadditivepositive valuationsweights withsumming demandto queries,one. improvingBoth ratios are independent of $m$. The max-min result improves the $2n-1$ bound of Khot and Ponnuswami. For additive valuations and for submodular valuations with value(2007). queries,We wealso give simpler $O(\sqrt n)$-approximation algorithms. Forfor additive valuations, thisand improvesfor submodular valuations with value queries, improving the $O(\sqrt{n\log n})$ bound of Saha and Srinivasan. For these two classes, $O(m^\varepsilon2018)$-approximation algorithms are also known forin everythe constantadditive $\varepsilon>0$case. Our ratios dependFor onlyNash onwelfare, $nΓ(w)=1$ and match, upfor toequal constants,weights theand $ΩΓ(w)\le\sqrt n)$ integralityin gapgeneral. ofWe theprove configurationa configuration-LP. Forintegrality thegap weightedof Nash$Θ(Γ(w))$ socialfor welfareevery problemprescribed withweight subadditivevector, even for XOS valuations. andFor demandevery queriesfixed $D>0$, we givealso anprove $O(Γ(w))$information-theoretic approximation algorithm,lower wherebounds of $ΓΩ_D(w)=\exp\bigl((sqrt{n/\ln n-H(w))/2\bigr})$ and $HΩ_D(Γ(w)/\sqrt{\ln n})$, isrespectively, thefor entropyrandomized ofalgorithms themaking weightat vectormost $wn^D$ adaptive queries. TheThese factorhold on XOS instances with $Γm=Θ(wn)$ equalsitems, $1$even forwith equalsimultaneous weightsaccess to value, demand, and isXOS atsupporting-clause mostoracles. $\sqrtThey n$apply forboth allto weights.guarantees Wewith showsuccess thatprobability thisat ratioleast is$2/3$ tightand forto algorithmsguarantees analyzedin againstexpected thewelfare. configurationThe LP:weighted bound holds for every prescribed weight vector, the configuration LP has anand integralityis gapvacuous ofbelow $Θ(Γ(w))$,one. evenThus forthe XOSoracle valuations,lower andbounds thisare gapwithin isan $ΩO(\sqrt{\ln n})$ whenfactor oneof agentour hasupper almostbounds, allwhile the weightconfiguration-LP gaps match them up to constants. ForOur submodularalgorithms valuations,build on random matchings for large items and charge the configurationaggregate LPloss hasacross aconfigurations constantto integralitythe gapat formost all$n$ weightsmatched items.