Distributed Algorithms for Matching in Hypergraphs

TL;DR

This paper introduces the first parallel algorithms for maximum matching in $d$-uniform hypergraphs within the MPC model, achieving various approximation ratios with efficient resource usage.

Contribution

It presents three novel parallel algorithms for hypergraph matching in the MPC model, analyzing their efficiency and approximation guarantees.

Findings

An $O( ext{log} n)$-round $d$-approximation algorithm with $O(nd)$ space.

A 3-round $O(d^2)$-approximation algorithm with $ ilde{O}( ext{sqrt}(nm))$ space.

A 3-round algorithm for subgraphs with approximation ratio $(d-1+rac{1}{d})^2$ using $ ilde{O}( ext{sqrt}(nm))$ space.

Abstract

$ $We study the $d$-Uniform Hypergraph Matching ($d$-UHM) problem: given an $n$-vertex hypergraph $G$ where every hyperedge is of size $d$, find a maximum cardinality set of disjoint hyperedges. For $d\geq3$, the problem of finding the maximum matching is NP-complete, and was one of Karp's 21 $\mathcal{NP}$-complete problems. In this paper we are interested in the problem of finding matchings in hypergraphs in the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In this model, we present the first three parallel algorithms for $d$-Uniform Hypergraph Matching, and we analyse them in terms of resources such as memory usage, rounds of communication needed, and approximation ratio. The highlights include: $\bullet$ A $O(\log n)$-round $d$-approximation algorithm that uses $O(nd)$ space per machine. $\bullet$ A $3$-round, $O(d^2)$-approximation algorithm that uses $\tilde{O}(\sqrt{nm})$ space per machine. $\bullet$ A $3$-round algorithm that computes a subgraph containing a $(d-1+\frac{1}{d})^2$-approximation, using $\tilde{O}(\sqrt{nm})$ space per machine for linear hypergraphs, and $\tilde{O}(n\sqrt{nm})$ in general.