The Complexity of Partition Functions on Hermitian Matrices

Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants, for example, the number of independent sets of a graph or the number nowhere zero k-flows. Contributing to recent developments on the complexity of partition functions we study the complexity of partition functions with complex values. These functions are usually determined by a square matrix A and it was shown by Goldberg, Grohe, Jerrum, and Thurley that for each real-valued symmetric matrix, the corresponding partition function is either polynomial time computable or #P-hard. Extending this result, we give a complete description of the complexity of partition functions definable by Hermitian matrices. These can also be classified into polynomial time computable and #P-hard ones. Although the criterion for polynomial time computability is not describable in a single line, we give a clear account of it in terms of structures associated with Abelian groups.

[1]  Libor Barto,et al.  Constraint Satisfaction Problems of Bounded Width , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[2]  L. Lovasz,et al.  Reflection positivity, rank connectivity, and homomorphism of graphs , 2004, math/0404468.

[3]  Alexander Schrijver,et al.  Graph invariants in the spin model , 2009, J. Comb. Theory, Ser. B.

[4]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[5]  Pierre de la Harpe,et al.  Graph Invariants Related to Statistical Mechanical Models: Examples and Problems , 1993, J. Comb. Theory, Ser. B.

[6]  Martin E. Dyer,et al.  A Complexity Dichotomy For Hypergraph Partition Functions , 2010, computational complexity.

[7]  Leslie Ann Goldberg,et al.  A Complexity Dichotomy for Partition Functions with Mixed Signs , 2008, SIAM J. Comput..

[8]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[9]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP , 2009, SIAM J. Comput..

[10]  Janos Simon,et al.  On the Difference Between One and Many (Preliminary Version) , 1977, ICALP.

[11]  Martin E. Dyer,et al.  On the complexity of #CSP , 2010, STOC '10.

[12]  Jin-Yi Cai,et al.  A Decidable Dichotomy Theorem on Directed Graph Homomorphisms with Non-negative Weights , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[13]  Martin E. Dyer,et al.  The complexity of approximating bounded-degree Boolean #CSP , 2012, Inf. Comput..

[14]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[15]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[16]  Alexander Schrijver,et al.  Graph parameters and semigroup functions , 2008, Eur. J. Comb..

[17]  D. Welsh,et al.  On the computational complexity of the Jones and Tutte polynomials , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Daniel Richardson.  Multiplicative independence of algebraic numbers and expressions , 2001 .

[19]  Andrei A. Bulatov,et al.  The complexity of partition functions , 2005, Theor. Comput. Sci..

[20]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

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

[22]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[23]  Martin E. Dyer,et al.  The Complexity of Weighted Boolean #CSP with Mixed Signs , 2009, Theor. Comput. Sci..

[24]  Rudolf Mathon,et al.  A Note on the Graph Isomorphism counting Problem , 1979, Inf. Process. Lett..

[25]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[26]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[27]  Martin E. Dyer,et al.  The complexity of counting graph homomorphisms , 2000, Random Struct. Algorithms.

[28]  N. Linial.  Hard enumeration problems in geometry and combinatorics , 1986 .

[29]  Martin E. Dyer,et al.  On counting homomorphisms to directed acyclic graphs , 2006, JACM.

[30]  L. Lovász.  Operations with structures , 1967 .

[31]  Jin-Yi Cai,et al.  Graph Homomorphisms with Complex Values: A Dichotomy Theorem , 2009, SIAM J. Comput..

[32]  László Lovász,et al.  The rank of connection matrices and the dimension of graph algebras , 2004, Eur. J. Comb..