The Submodular Santa Claus Problem

TL;DR

This paper extends approximation algorithms for the Santa Claus problem to cases where players have monotone submodular valuation functions, achieving better ratios using a novel LP relaxation.

Contribution

It introduces an improved approximation algorithm for the Santa Claus problem with submodular valuations, surpassing previous bounds.

Findings

Achieved an $O(n^{ ext{epsilon}})$-approximation for submodular valuations.

Developed a recursive LP relaxation with a block structure.

Improved upon previous approximation ratios for the problem.

Abstract

We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^{\epsilon})$-approximation algorithm with polynomial running time for any constant $\epsilon > 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.