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Submodular Max-Min Allocation Under Identical Valuations

Author Kimon Boehmer

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Subject Classification

ACM Subject Classification
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Linear programming
  • Theory of computation → Packing and covering problems
Keywords
  • Submodularity
  • Approximation algorithms
  • Allocation
  • Configuration LP

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Abstract

In the problem of Submodular Max-Min Allocation, we are given a set of items, a set of players, and monotone submodular valuation functions that represent the satisfaction of a player with a certain subset of items. The goal is to find an allocation of the items to the players that maximizes the lowest satisfaction among all players. 
We study this problem in the special case where all players have the same valuation function. We devise a greedy algorithm which gives a 0.4-approximation, improving the previously best factor of 10/27 ≈ 0.37 by Uziahu and Feige.
Furthermore, we study the integrality gap of the configuration LP when players have identical valuations. By constructing a variable assignment to the dual from a primal integral solution, we give the first constant upper bound on the integrality gap for submodular valuations. Generalizing the result to the case where players' allocations must be independent in k given matroids, we derive a 𝒪(k)-estimation algorithm for max-min allocation subject to k matroid constraints under identical valuations.

Cite As Get BibTex

Kimon Boehmer. Submodular Max-Min Allocation Under Identical Valuations. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.8

Author Details

Kimon Boehmer
  • DIENS, École normale supérieure, PSL University, Paris, France
  • LIP, ENS Lyon, France

Acknowledgements

I want to thank Chien-Chung Huang for introducing me to the topic and for many helpful dicussions. Also, I thank the anonymous reviewers for their valuable suggestions.

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