Nearly-Perfect Hypergraph Packing is in NC

Answering a question of RR odl and Thoma, we show that the Nibble Algorithm for nding a collection of disjoint edges covering almost all vertices in an almost regular, uniform hypergraph with negligible pair degrees can be derandomized and parallelized to run in polylog time on polynomially many parallel processors. In other words, the nearly-perfect packing problem on this class of hypergraphs is in the complexity class NC. Finding a collection of disjoint edges in an r-uniform hypergraph (a packing or matching) containing a speciied number of edges or showing that none exists is known to be NP-complete for r 3 8] despite being signiicantly easier for graphs (r = 2), where the problem is in RNC 10]. On the other hand, it was shown by Frankl and RR odl 5] and Pippenger and Spencer 13] that if the hypergraph is suuciently regular and has negligible pair degrees, it must contain a packing covering most of the vertices. (We state this more precisely below.) In 16] RR odl and Thoma ask whether the problem of nding such a nearly-perfect packing in a hypergraph from this class is in NC. It isn't too diicult to see that the proofs of 5] and 13] give RNC-class algorithms, but it is not obvious that they can be derandomized to produce NC-class algorithms. The aim of this note is to show how this may be done. We start with some deenitions. An r-uniform hypergraph H is a set V (H), whose elements are called vertices, and a set E(H) of r element subsets of V (H), whose elements are called edges. The number of vertices is usually denoted by n. An "-near-perfect packing is a collection of pair-wise disjoint edges with the property that at least (1 ? ")n of the vertices are contained in some edge of the packing. For vertices x and y, deg(x) and deg(x; y) denote respectively the number of edges containing x and the number of edges containing both x and y. We denote by (1) some quantity between (1 ?) and (1 +).