Computational complexity of counting problems on 3-regular planar graphs

Abstract A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial interpolation technique in high dimension to prove the #P-completeness of problems on graphs with special requirements. Using this method, we show that #3-Regular Bipartite Planar Vertex Covers is #P-complete. Furthermore, we use Valiant’s Holant Theorem to construct a holographic reduction from it to #2,3-Regular Bipartite Planar Matchings, establishing the #P-completeness of the latter. Finally, we completely classify the problems #Planar Read-twice 3SAT with different ternary symmetric relations according to their computational complexity, by giving several more applications of holographic reduction in proving the #P-completeness of the corresponding counting problems.

[1]  Jin-Yi Cai,et al.  Valiant's Holant Theorem and matchgate tensors , 2007, Theor. Comput. Sci..

[2]  P. W. Kasteleyn.  The Statistics of Dimers on a Lattice , 1961 .

[3]  Jehoshua Bruck,et al.  Networks of relations , 2005 .

[4]  M. Fisher,et al.  Dimer problem in statistical mechanics-an exact result , 1961 .

[5]  Martin E. Dyer,et al.  Graph orientations with no sink and an approximation for a hard case of #SAT , 1997, SODA '97.

[6]  Leslie G. Valiant,et al.  Accidental Algorithms , 2006, FOCS.

[7]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .

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

[9]  Leslie G. Valiant,et al.  Holographic Algorithms (Extended Abstract) , 2004, FOCS.

[10]  Mingji Xia,et al.  #3-Regular Bipartite Planar Vertex Cover is #P-Complete , 2006, TAMC.

[11]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[12]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

[13]  Jin-Yi Cai,et al.  Some Results on Matchgates and Holographic Algorithms , 2007, Int. J. Softw. Informatics.

[14]  M. Fisher.  Statistical Mechanics of Dimers on a Plane Lattice , 1961 .

[15]  Maciej Liskiewicz,et al.  The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes , 2003, Theor. Comput. Sci..

[16]  Jin-Yi Cai,et al.  Valiant's Holant Theorem and Matchgate Tensors , 2006, TAMC.

[17]  G. Spagnolo,et al.  Networks of Relations , 2004 .

[18]  Harry B. Hunt,et al.  The Complexity of Planar Counting Problems , 1998, SIAM J. Comput..

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

[20]  G. Sposito,et al.  Graph theory and theoretical physics , 1969 .