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. Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) 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 (Proceedings of the 38th ACM Symposium on Theory of Computing, 2006). 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 (Graphs Comb 11(3):245–248, 1995) for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods, to replace the combinatorial argument of Haxell (Graphs Comb 11(3):245–248, 1995) for the restricted max–min allocation problem. 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. Our approach applies a criterion of Aharoni and Haxell, and Meshulam, for the existence of independent transversals in graphs, which involves the connectedness of the independence complex. This is complemented by a graph process of Meshulam that decreases the connectedness of the independence complex in a controlled fashion and hence, tailored appropriately to the problem, can verify the criterion. In our applications we aim to establish the flexibility of the approach and hence argue for it to be a potential asset in other optimization problems involving hypergraph matchings.