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Title: Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs
In an instance of the weighted Nash Social Welfare problem, we are given a set of m indivisible items, G, and n agents, A, where each agent i in A has a valuation v_ij ≥ 0 for each item j in G. In addition, every agent i has a non-negative weight w_i such that the weights collectively sum up to 1. The goal is to find an assignment of items to players that maximizes the weighted geometric mean of the valuation received by the players. When all the weights are equal, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present an approximation algorithm whose approximation depends on the KL-divergence between the weight distribution and the uniform distribution. We generalize the convex programming relaxations for the symmetric variant of Nash Social Welfare presented in [CDG+17, AGSS17] to two different mathematical programs. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation.  more » « less
Award ID(s):
1910423 2106444
NSF-PAR ID:
10473237
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
ACM
Date Published:
Journal Name:
Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms
Format(s):
Medium: X
Location:
Alexandria, Virginia
Sponsoring Org:
National Science Foundation
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