Counting Homomorphisms via Hypergraph-Based Structural Restrictions [chapter]

Tommy Färnqvist
2012 Lecture Notes in Computer Science  
The way in which the graph structure of the constraints influences the computational complexity of counting constraint satisfaction problems (#CSPs) is well understood for constraints of bounded arity. The situation is less clear if there is no bound on the arities. Here we initiate the systematic study of these problems and identify new classes of polynomial time solvable instances based on dynamic programming over tree decompositions, in a way generalizing wellknown approaches to
more » ... optimization problems on bounded treewidth graphs, but basing the decompositions on various hypergraph width measures from the literature on plain CSPs. A large class of problems in different areas of computer science can be viewed as constraint satisfaction problems (CSPs). This includes problems in artificial intelligence, database theory, scheduling, frequency assignment, graph theory, and satisfiability. In this paper we study the problem of determining how many solutions there are to a given CSP instance. Our ability to solve this problem has several applications in artificial intelligence, statistical physics, and more recently in guiding backtrack search heuristics to find solutions to CSPs [20] . Of course, the problem is #P-complete in general. Feder and Vardi [9] observed that constraint satisfaction problems can be described as homomorphism problems for relational structures. For every two classes of relational structures C, D, let #HOM(C, D) be the problem of counting the number of homomorphisms from a structure A ∈ C to a given arbitrary structure B ∈ D. To simplify the notation, if either C or D is the class of all structures, we just use the placeholder '_'. Flum and Grohe [10] and Dalmau and Jonsson [6] have studied so called structural restrictions, i.e. the question of how to restrict C, so that #HOM(C, _) is polynomial-time solvable. They prove the following: Assume that FPT = #W[1]. Then for every recursively enumerable class C of structures of bounded arity, #HOM(C, _) is polynomial-time solvable if and only if every structure in C has treewidth at most w (for some fixed w).
doi:10.1007/978-3-642-32147-4_34 fatcat:ir6ffm3qzzecfo4qrudun7u6me