Approximate Counting of Matchings in $(3,3)$-Hypergraphs
  We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as $(3,3)$-hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite $(3,3)$-hypergraphs. The proof technique of this paper uses the general correlation decay technique and a new combinatorial analysis of the underlying structures of the intersection graphs. The proof method could be also of independent interest. 
Andrzej Dudek, Marek Karpinski, Andrzej Ruci\'nski, and Edyta   Szyma\'nska
math.CO cs.DS