13.12.2010

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k − 1)-wise) vertex degree δ(H) at least c|V(H)| is NP-complete for c < 1/k and trivial for c ≥ 1/2 leaving the status of the problem with c in the interval [1/k,1/2) widely open. In this paper we show, somehow surprisingly, that 1/2 is not the threshold for 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(H) ≥ (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.