Combinatorial Algorithm for Restricted Max-Min Fair Allocation

TL;DR

This paper presents a new combinatorial 13-approximation algorithm for the restricted max-min fair allocation problem, narrowing the gap between estimation and approximation guarantees using advanced local search techniques.

Contribution

It introduces a purely combinatorial 13-approximation algorithm for the problem, improving upon previous methods and employing novel ideas like lazy updates and greedy players.

Findings

Achieved a 13-approximation guarantee for the problem.

Developed a polynomial-time local search algorithm with new techniques.

Provided analysis showing the algorithm's effectiveness using Configuration-LP only in analysis.

Abstract

We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain Configuration-LP can be used to estimate the value of the optimal allocation to within a factor of $4 + {\epsilon}$. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee. In this paper we significantly narrow this gap by giving a $13$-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [Hax95] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the Configuration-LP only in the analysis.