Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole

In the max-min allocation problem a set $P$ of players are to be allocated disjoint subsets of a set $R$ of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bez\'akov\'a and Dani showed that this problem is NP-hard to approximate within a factor less than $2$, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods for the restricted max-min allocation problem, to replace the combinatorial argument. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of $3.808$ to $3.534$. We also study the $(1,\varepsilon)$-restricted version, in which resources can take only two values, and improve the integrality gap in most cases.