29.09.2009

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 &delta at least c|V(H)| for c < 1/k and trivial for c ≥ 1/2 leaving the status of the problems with c in the interval [1/k,1/2) widely open. In this paper we show, somehow surprisingly, that 1/2, 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 (1/2 - ε)|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.