Towards a dichotomy theorem for the counting constraint satisfaction problem

The Counting Constraint Satisfaction Problem (#CSP) over a finite domain can be expressed as follows: given a first-order formula consisting of a conjunction of predicates, determine the number of satisfying assignments to the formula. #CSP can be parametrized by the set of allowed constraint predicates. In this paper we start a systematic study of subclasses of #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of #CSP which are tractable, i.e. solvable in polynomial time, from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision CSP. Then we prove that if a subclass of the #CSP is tractable, then constraints allowed by the class satisfy some very restrictive condition: it has to have a Mal'tsev polymorphism, that is a ternary operation m(x, y, z) such that m(x, y, y) = m(y, y, x) = x. This condition uniformly explains all existing complexity results for particular cases of #CSP, and allows us to obtain new results and to conjecture a criterion distinguishing tractable counting CSPs. We also obtain a dichotomy theorem for the complexity of #CSP with a 3-element domain and give new simpler proofs of the dichotomy results for the problem of counting graph homomorphisms.