A complexity dichotomy for partition functions with mixed signs

TL;DR

This paper provides a complete classification of the computational complexity of partition functions, showing they are either polynomial-time computable or #P-complete, with a polynomial-time decision procedure based on matrix properties.

Contribution

It establishes a dichotomy theorem for partition functions with mixed signs and offers a simple algebraic criterion for tractability in cases described by Hadamard matrices.

Findings

Partition functions are either in P or #P-complete.

Decidability of complexity class based on matrix analysis is polynomial.

Tractable cases for Hadamard matrices are characterized by quadratic polynomials over GF(2).

Abstract

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition functions of certain "spin glass" models of statistical physics such as the Ising model. Building on earlier work by Dyer, Greenhill and Bulatov, Grohe, we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices -- these turn out to be central in our proofs -- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those "representable" by a quadratic polynomial over the field GF(2).