Santa Claus Meets Hypergraph Matchings

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We consider the problem of max-min fair allocation of indivisible goods. Our focus will be on the restricted version of the problem in which there are m items, each of which associated with a non-negative value. There are also n players and each player is only interested in some of the items. The goal is to distribute the items between the players such that the least happy person is as happy as possible, i.e. one wants to maximize the minimum of the sum of the values of the items given to any player. This problem is also known as the Santa Claus problem [3]. Feige [9] proves that the integrality gap of a certain configuration LP, described by Bansal and Sviridenko [3], is bounded from below by some (unspecified) constant. This gives an efficient way to estimate the optimum value of the problem within a constant factor. However, the proof in [9] is nonconstructive: it uses the Lovasz local lemma and does not provide a polynomial time algorithm for finding an allocation. In this paper, we take a different approach to this problem, based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is bounded by \(\frac{1}{5}\) . Our proof is nonconstructive in the following sense: it does provide a local search algorithm which finds the corresponding allocation, but this algorithm is not known to converge to a local optimum in a polynomial number of steps.