Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation

TL;DR

This paper presents a quasi-polynomial time local search algorithm for the restricted max-min fair allocation problem, improving the running time from exponential to quasi-polynomial while maintaining strong approximation guarantees.

Contribution

It introduces a novel modification and analysis of local search that achieves quasi-polynomial runtime without solving the configuration LP directly.

Findings

Achieves a quasi-polynomial approximation algorithm with a performance guarantee better than previous methods.

Significantly improves the running time of local search from exponential to quasi-polynomial.

Provides a purely combinatorial algorithm that does not require solving the configuration LP.

Abstract

The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, Asadpour et al. proved that a certain configuration LP can be used to estimate the optimal value within a factor ${1}/{(4+\epsilon)}$, for any $\epsilon>0$, but at the same time it is not known how to efficiently find a solution with a comparable performance guarantee. A natural question that arises from their work is if the difference between these guarantees is inherent or because of a lack of suitable techniques. We address this problem by giving a quasi-polynomial approximation algorithm with the mentioned performance guarantee. More specifically, we modify the local search of Asadpour et al. and provide a novel analysis that lets us significantly improve the bound on its running time: from $2^{O(n)}$ to $n^{O(\log n)}$. Our techniques also have the interesting property that although we use the rather complex configuration LP in the analysis, we never actually solve it and therefore the resulting algorithm is purely combinatorial.