DeepAI

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

We consider hypergraphs on vertices P∪ 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/α)-fraction of its intersection with R (thus relaxing the matching constraint). Here α 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 α = 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 α = O(loglog(|R|)), when all hyperedges are sufficiently large (a condition that is necessary). In particular, this implies an O(loglog(|R|))-approximation algorithm for the Combinatorial Santa Claus problem with large hyperedges.

• 8 publications
• 4 publications
• 19 publications
08/30/2018

### Improved approximation algorithms for hitting 3-vertex paths

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### Approximation algorithms for hitting subgraphs

Let H be a fixed undirected graph on k vertices. The H-hitting set probl...
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### Combinatorial Bernoulli Factories: Matchings, Flows and Other Polytopes

A Bernoulli factory is an algorithmic procedure for exact sampling of ce...
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### A Perfect Sampler for Hypergraph Independent Sets

The problem of uniformly sampling hypergraph independent sets is revisit...
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### Linearly ordered colourings of hypergraphs

A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns a...
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### Hypergraph regularity and random sampling

Suppose a k-uniform hypergraph H that satisfies a certain regularity ins...
09/21/2020

### Distributed Algorithms for Matching in Hypergraphs

We study the d-Uniform Hypergraph Matching (d-UHM) problem: given an n-v...