A complexity trichotomy for approximately counting list H-colourings

TL;DR

This paper classifies the computational complexity of approximately counting list H-colourings based on the structure of graph H, revealing a natural trichotomy that includes polynomial-time solvable, #BIS-equivalent, and #P-complete cases.

Contribution

It introduces a natural graph-theoretic trichotomy for counting list H-colourings, extending hardness results to bounded-degree graphs, and provides a largely self-contained proof without universal algebra.

Findings

Polynomial-time cases for certain graph classes

Approximate counting is #BIS-equivalent for some classes

Hardness results apply to graphs with maximum degree 6

Abstract

We examine the computational complexity of approximately counting the list H-colourings 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-colourings 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-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem which 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-colourings 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 (i) it has a natural formulation in terms of hereditary graph classes, and (ii) 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.