A Complexity Trichotomy for Approximately Counting List H-Colorings

We examine the computational complexity of approximately counting the list H-colorings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph, then counting list H-colorings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph, then approximately counting list H-colorings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem that is believed to be of intermediate complexity—it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colorings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (1) it has a natural formulation in terms of hereditary graph classes, and (2) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.