Counting List Matrix Partitions of Graphs

Given a symmetric DxD matrix M over {0, 1, *}, a list M-partition of a graph G is a partition of the vertices of G into D parts which are associated with the rows of M. The part of each vertex is chosen from a given list in such a way that no edge of G is mapped to a 0 in M and no non-edge of G is mapped to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions including graph colourings, split graphs and homogeneous sets, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. This paper focuses on the problem of counting list M-partitions, given a graph G and given lists for each vertex of G. We give an algorithm that solves this problem in polynomial time for every (fixed) matrix M for which the problem is tractable. The algorithm relies on data structures such as sparse-dense partitions and sub cube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of M in which the interactions of 0s and 1s is controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (#CSPs) which we show how to solve using a constraint satisfaction technique known as \"arc-consistency\". For every matrix M for which our algorithm fails, we show that the problem of counting list M-partitions is #P-complete. Furthermore, we give an explicit characterisation of the dichotomy theorem - counting list M-partitions is tractable (in FP) if and only if the matrix M has a structure called a derectangularising sequence. Finally, we show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete.