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.

[1]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[2]  Paul D. Seymour,et al.  The roots of the independence polynomial of a clawfree graph , 2007, J. Comb. Theory B.

[3]  R. Dobrushin.  Prescribing a System of Random Variables by Conditional Distributions , 1970 .

[4]  Fadnavis.  Approximating independence polynomials of claw-free graphs Sukhada , 2011 .

[5]  Catherine S. Greenhill.  The complexity of counting colourings and independent sets in sparse graphs and hypergraphs , 2000, computational complexity.

[6]  M. Luby,et al.  Fast convergence of the Glauber dynamics for sampling independent sets , 1999 .

[7]  O. J. Heilmann.  Existence of phase transitions in certain lattice gases with repulsive potential , 1972 .

[8]  Martin E. Dyer,et al.  On Counting Independent Sets in Sparse Graphs , 2002, SIAM J. Comput..

[9]  David Gamarnik,et al.  Simple deterministic approximation algorithms for counting matchings , 2007, STOC '07.

[10]  Allan Sly,et al.  The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[11]  L. Beineke.  Characterizations of derived graphs , 1970 .

[12]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[13]  Dror Weitz,et al.  Counting independent sets up to the tree threshold , 2006, STOC '06.

[14]  Allan Sly,et al.  Computational Transition at the Uniqueness Threshold , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[15]  Marek Karpinski,et al.  Approximate Counting of Matchings in Sparse Uniform Hypergraphs , 2012, ANALCO.

[16]  Frank Kelly,et al.  Stochastic Models of Computer Communication Systems , 1985 .