Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

Abstract

We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with factor independent of m is known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations.In this paper, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations respectively. Both algorithms are simple to understand and involve nontrivial modifications of a greedy repeated matchings approach. Allocations of high valued items are done separately by un-matching certain items and re-matching them, by processes that are different in both algorithms. We show that these approaches achieve approximation factors of O(n) and O(n log n) for additive and submodular case respectively, which is independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1).Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an e e−1 factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of e e−1 , hence resolving it completely.

Approximating nash social welfare under rado valuations

“…The NSW problem is NP-hard even for two identical agents with additive valuations: the partition problem reduces to the NSW problem [56]. Moreover, the problem is NP-hard to approximate within a factor better than 1.069 for additive valuations [27], and better than 1.5819 for submodular valuations [30]. These results hold already in the symmetric case.…”

“…Beyond 'additive-like' valuations or the asymmetric NSW problem no constant-factor approximation algorithms are known. Here, the state-of-the-art are O(n)-approximation algorithms for the asymmetric Nash problem under subadditive valuations [9,16,30]. However, no better than O(n) approximation has been achieved even for special cases such as OXS valuations, or only two types of agents with weights 1 or 2 under additive valuations.…”

“…This can be efficiently solved as a maximum weight matching problem. The algorithm in [30] also starts with such a matching. One cannot commit to assigning these items to the agents, as it may result in an arbitrarily bad outcome; the approach in [30] is an intricate combinatorial scheme with iterated matchings and reallocations to obtain an O(n log n) approximation for submodular valuations.…”

“…The algorithm in [30] also starts with such a matching. One cannot commit to assigning these items to the agents, as it may result in an arbitrarily bad outcome; the approach in [30] is an intricate combinatorial scheme with iterated matchings and reallocations to obtain an O(n log n) approximation for submodular valuations. Our result implies that the mixed integer relaxation that requires H to be integrally allocated has a constant integrality gap, in contrast to the standard continuous relaxation.…”

EFX Exists for Three Agents

“…Therefore, the Nash welfare of an allocation is also considered as a measure of efficiency and fairness of an allocation. However, finding an allocation with the maximum Nash welfare is APX-hard [25], and its approximation has received a lot of attention recently, e.g., [2,3,5,11,[14][15][16][17]. Barman et al [5] give a pseudopolynomial algorithm to find an allocation that is both EF1 and Pareto-optimal.…”