The Combinatorial Santa Claus Problem or: How to Find Good Matchings in Non-Uniform Hypergraphs

TL;DR

This paper introduces the Combinatorial Santa Claus problem, a relaxed matching problem in hypergraphs, and shows that non-uniform regular hypergraphs admit approximate solutions within a double logarithmic factor, leading to improved approximation algorithms.

Contribution

It proves that non-uniform regular hypergraphs have relaxed matchings with an approximation factor of O(log log |R|), extending previous uniform hypergraph results.

Findings

Non-uniform regular hypergraphs admit relaxed matchings with O(log log |R|) approximation.

This result leads to an O(log log |R|)-approximation algorithm for the problem.

The problem is shown to be nearly equivalent to the Santa Claus problem in terms of approximability.

Abstract

We consider hypergraphs on vertices $P\cup R$ where each hyperedge contains exactly one vertex in $P$. Our goal is to select a matching that covers all of $P$, but we allow each selected hyperedge to drop all but an $(1/\alpha)$-fraction of its intersection with $R$ (thus relaxing the matching constraint). Here $\alpha$ is to be minimized. We dub this problem the Combinatorial Santa Claus problem, since we show in this paper that this problem and the Santa Claus problem are almost equivalent in terms of their approximability. The non-trivial observation that any uniform regular hypergraph admits a relaxed matching for $\alpha = O(1)$ was a major step in obtaining a constant approximation rate for a special case of the Santa Claus problem, which received great attention in literature. It is natural to ask if the uniformity condition can be omitted. Our main result is that every (non-uniform) regular hypergraph admits a relaxed matching for $\alpha = O(\log\log(|R|))$, when all hyperedges are sufficiently large (a condition that is necessary). In particular, this implies an $O(\log\log(|R|))$-approximation algorithm for the Combinatorial Santa Claus problem with large hyperedges.