Counting Homomorphisms and Partition Functions

Homomorphisms between relational structures are not only fundamental mathematical objects, but are also of great importance in an applied computational context. Indeed, constraint satisfaction problems, a wide class of algorithmic problems that occur in many different areas of computer science such as artificial intelligence or database theory, may be viewed as asking for homomorphisms between two relational structures [FV98]. In a logical setting, homomorphisms may be viewed as witnesses for positive primitive formulas in a relational language. As we shall see below, homomorphisms, or more precisely the numbers of homomorphisms between two structures, are also related to a fundamental computational problem of statistical physics. Homomorphisms of graphs are generalizations of colorings, and for that reason a homomorphism from a graph G to a graph H is also called an H-coloring of G. Note that if H is the complete graph on k-vertices, then an H-coloring of a graph G may be viewed as a proper k-coloring of G in the usual graph theoretic sense that adjacent vertices are not allowed to get the same color. It is thus no surprise that the computational complexity of various algorithmic problems related to homomorphisms, in particular the decision problem of whether a homomorphism between two given structures exists and the counting problem of determining the number of such homomorphisms, have been intensely studied. (For the decision problem, see, for example, [BKN09, Bul06, BKJ05, Gro07, HN90]. References for the counting problem will be given in Section 3. Other related problems, such as optimization or enumeration problems, have been studied, for example, in [Aus07, BDGM09, DJKK08, Rag08, SS07].) In this article, we are concerned with the complexity of counting homomorphisms from a given structure A to a fixed structure B. Actually, we are mainly interested in a generalization of this problem to be introduced in the next section. We almost exclusively focus on graphs. The first part of the article, consisting of the following two sections, is a short survey of what is known about the problem. In the second part, consisting of the remaining Sections 4-9, we give a proof of a theorem due to Bulatov and the first author of this paper [BG05], which classifies the complexity of partition functions described by matrices with non-negative entries. The proof we give here is essentially the same as the original one, with a few shortcuts due to [Thu09], but it is phrased in a different, more graph theoretical language that may make it more accessible to most readers.