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).
We study the complexity of a family of graph invariants known as partition functions or homomorphism functions (see, for example, [14,21,22]). Many natural graph invariants can be expressed as homomorphism functions, among them the number of k-colourings, the number of independent sets, and the number of nowhere-zero k-flows of a graph. The functions also appear as the partition functions of certain "spin-glass" models of statistical physics such as the Ising model or the q-state Potts model.
Let A ∈ R m×m be a symmetric real matrix with entries A i,j . The partition function Z A associates with every graph G = (V, E) the real number
We refer to the row and column indices of the matrix, which are elements of [m] := {1, . . . , m}, as spins. We use the term configuration to refer to a mapping ξ : V → [m] assigning a spin to each vertex of the graph. To avoid difficulties with models of real number computation, throughtout this paper we restrict our attention to algebraic numbers. Let R A denote the set of algebraic real numbers. 1Our main result is a dichotomy theorem stating that for every symmetric matrix A ∈ R m×m A the partition function Z A is either computable in polynomial time or #P-hard. This extends earlier results by Dyer and Greenhill [9], who proved the dichotomy for 0-1matrices, and Bulatov and Grohe [6], who proved it for nonnegative matrices. Therefore, in this paper we are mainly interested in matrices with negative entries.
In the following, let G = (V, E) be a graph with N vertices. Consider the matrices
It is not hard to see that Z S (G) is the number of independent sets of a graph G and Z C 3 (G) is the number of 3-colourings of G. More generally, if A is the adjacency matrix of a graph H then Z A (G) is the number of homomorphisms from G to H. Here we allow H to have loops and parallel edges; the entry A i,j in the adjacency matrix is the number of edges from vertex i to vertex j.
Let us turn to matrices with negative entries. Consider
Then 1 2 Z H 2 (G) + 2 N -1 is the number of induced subgraphs of G with an even number of edges. Hence up to a simple transformation, Z H 2 counts induced subgraphs with an even number of edges. To see this, observe that for every configuration ξ : V → [2] the term {u,v}∈E A ξ(u),ξ(v) is 1 if the subgraph of G induced by ξ -1 (2) has an even number of edges and -1 otherwise. Note that H 2 is the simplest nontrivial Hadamard matrix. Hadamard matrices will play a central role in this paper. Another simple example is the matrix
It is a nice exercise to verify that for connected G the number Z U (G) is 2 N if G is Eulerian and 0 otherwise. A less obvious example of a counting function that can be expressed in terms of a partition function is the number of nowhere-zero k-flows of a graph. It can be shown that the number of nowhere-zero k-flows of a graph G with N vertices is k -N • Z F k (G), where F k is the k × k matrix with (k -1)s on the diagonal and -1s everywhere else. This is a special case of a more general connection between partition functions for matrices A with diagonal entries d and off diagonal entries c and certain values of the Tutte polynomial. This well-known connection can be derived by establishing certain contraction-deletion identities for the partition functions. For example, it follows from [24,Equations (3.5.4)] and [23,Equation (2.26) and (2.9)]
Like the complexity of graph polynomials [2,16,18,20] and constraint satisfaction problems [1,3,4,5,12,15,17], which are both closely related to our partition functions, the complexity of partition functions has already received quite a bit of a attention. Dyer and Greenhill [9] studied the complexity of counting homomorphisms from a given graph G to a fixed graph H without parallel edges. (Homomorphisms from G to H are also known as H-colourings of G.) They proved that the problem is in polynomial time if every connected component of H is either a complete graph with a loop at every vertex or a complete bipartite graph, and the problem is #P-hard otherwise. Note that, in particular, this gives a complete classification of the complexity of computing Z A for symmetric 0-1-matrices A. Bulatov and Grohe [6] extended this to symmetric nonnegative matrices. To state the result, it is convenient to introduce the notion of a block of a matrix A. To define the blocks of A, it is best to view A as the adjacency matrix of a graph with weighted edges; then each non-bipartite connected component of this graph corresponds to one block and each bipartite connected component corresponds to two blocks. A formal definition will be given below. Bulatov and Grohe [6] proved that computing the function Z A is in polynomial time if the row rank of every block of A is 1 and #P -hard otherwise. The problem for matrices with negative entries was left open. In particular, Bulatov and Grohe asked for the complexity of the partition function Z H 2 for the matrix H 2 introduced in (1.1). Note that H 2 is a mat
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