Submodular Max-Min Allocation under Identical Valuations

TL;DR

This paper presents a new greedy algorithm for submodular max-min allocation with identical valuations, achieving a better approximation ratio and analyzing the integrality gap of the configuration LP.

Contribution

It introduces a 0.4-approximation greedy algorithm and provides the first constant upper bound on the integrality gap for this problem.

Findings

The greedy algorithm achieves a 0.4-approximation ratio.

The integrality gap of the configuration LP is bounded by a constant.

Extension to k-matroid constraints yields an O(k)-approximation algorithm.

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 $\frac{10}{27} \approx 0.37$ by Uziahu and Feige. Furthermore, we study the integrality gap of the \emph{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 $\mathcal{O}(k)$-estimation algorithm for max-min allocation subject to $k$ matroid constraints under identical valuations.