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In the max-min allocation problem a set $P$ of players are to be allocateddisjoint subsets of a set $R$ of indivisible resources, such that the minimumutility among all players is maximized. We study the restricted variant, alsoknown as the Santa Claus problem, where each resource has an intrinsic positivevalue, and each player covets a subset of the resources. Bez\'akov\'a and Danishowed 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 theConfiguration LP (CLP) of Bansal and Sviridenko. Accordingly, there has beenmuch interest in bounding the integrality gap of this CLP. The existingalgorithms and integrality gap estimations are all based one way or another onthe combinatorial augmenting tree argument of Haxell for finding perfectmatchings in certain hypergraphs. Our main innovation in this paper is tointroduce the use of topological methods for the restricted max-min allocationproblem, to replace the combinatorial argument. This approach yieldssubstantial improvements in the integrality gap of the CLP. In particular weimprove the previously best known bound of $3.808$ to $3.534$. We also studythe $(1,\varepsilon)$-restricted version, in which resources can take only twovalues, and improve the integrality gap in most cases.

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