The complexity of almost perfect matchings and other packing problems in uniform hypergraphs with high codegree

In this paper we prove that the problem of deciding whether a given k -uniform hypergraph H , with minimum ( k − 1 ) -wise vertex degree at least c | V ( H ) | , contains a matching missing exactly r vertices, that is, a set of disjoint edges of size ( | V ( H ) | − r ) / k , is NP-complete for c < 1 k , while for c > 1 k and r > 0 we provide a polynomial time algorithm for the corresponding search problem. For the perfect case, r = 0 , we show that the problem is NP-complete for c < 1 k and give a polynomial time algorithm for c > 1 2 leaving a hardness gap in ( 1 k , 1 2 ) . duction carries over to the more general case when, for 1 ≤ l ≤ k − 1 , the minimum l -wise codegree of a k -uniform hypergraph is considered. In particular, for k = 3 , l = 1 and r = 1 we deduce that the problem of deciding the existence of a perfect matching in a given k -uniform hypergraph H , with minimum degree at least c | V ( H ) | , is NP-complete for c < 5 9 , complementing a fact proved recently in Han et al. (2009) [8] that the problem is trivial for c > 5 9 . ition, we use our reduction to show that a problem of deciding the existence of a perfect packing of the cycle C 4 ( 3 ) into a 3-uniform hypergraph H with minimum 2 -wise vertex degree at least c | V ( H ) | is NP-complete only for c < 1 4 , which combined with a result from Kühn and Osthus (2006) [16] is, again, asymptotically tight.