Computer Science > Data Structures and Algorithms
[Submitted on 21 Sep 2020]
Title:Distributed Algorithms for Matching in Hypergraphs
View PDFAbstract:$ $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.
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