The paper presents an O(1)-approximation algorithm for the weighted Nash Social Welfare problem with additive valuations. The key highlights are:
The algorithm is based on solving a natural configuration LP for the problem, which has not been studied before. The configuration LP is solved to any desired precision using an approximate separation oracle.
The LP solution is then rounded using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems. The rounding algorithm maintains marginal probabilities and ensures that each agent gets at most one item from each group of items.
The analysis shows that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 (envy-free up to one item) allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most e^(1/e) by prior work.
The approximation ratio of e^(1/e) + ε matches the best known ratio for the unweighted case, improving upon the previous ratio of 5 * exp(2 * DKL(w || 1/n)) for the weighted case.
The authors believe the configuration LP approach could be useful in other settings, and leave as an open problem whether it can give an O(1)-approximation for the weighted Nash Social Welfare problem with submodular valuations.
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by Yuda Feng,Sh... a las arxiv.org 04-25-2024
https://arxiv.org/pdf/2404.15607.pdfConsultas más profundas