# The Complexity of Perfect Matching Problems on Dense Hypergraphs

## Abstract

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense *k*-uniform hypergraphs. Some of these problems are known to be notoriously hard. There is also a renewed interest recently in the very special cases of them. One of the goals of this paper is to shed some light on the tractability barriers for those problems.

It has been known that the perfect matching problems are NP-complete for the classes of hypergraphs *H* with *minimum* ((*k* − 1) −wise) *vertex degree* *δ* at least *c*|*V*(*H*)| for \(c<\frac 1k\) and trivial for \(c\ge\frac 12,\) leaving the status of the problems with *c* in the interval \([\frac 1k,\frac 12)\) widely open. In this paper we show, somehow surprisingly, that \(\frac 12\), in fact, is not a *threshold* for the tractability of the perfect matching problem, and prove the existence of an *ε*> 0 such that the perfect matching problem for the class of hypergraphs *H* with *δ* at least \((\frac 12-\epsilon)|V(H)|\) is solvable in *polynomial time*. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest in view of the known results for graphs.

## Keywords

Perfect Match Parallel Algorithm Polynomial Time Algorithm Parallel Complexity Auxiliary Graph## Preview

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