A complexity trichotomy for approximately counting list H-colourings

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.

💡 Research Summary

The paper investigates the computational complexity of approximately counting list H‑colourings of a graph, where a list H‑colouring is a homomorphism from an input graph G to a fixed target graph H that respects per‑vertex colour lists. While exact counting for #List‑H‑Col is already known to be either polynomial‑time solvable or #P‑complete (by Dyer and Greenhill), the authors show that the landscape for approximation is far richer and can be captured by a clean trichotomy based solely on the structural class of H.

The main theorem (Theorem 1) states that for any connected undirected graph H (loops allowed):

1. Polynomial‑time case – If H is an irreflexive complete bipartite graph or a reflexive complete graph, then #List‑H‑Col can be computed exactly in polynomial time, and consequently admits an FPRAS. This mirrors the exact‑counting dichotomy, because the colour choices at different vertices become independent.

2. #BIS‑equivalent case – If H is an irreflexive bipartite permutation graph or a reflexive proper interval graph (but not covered by case 1), then approximating #List‑H‑Col is AP‑equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. #BIS is believed to be of intermediate difficulty: no FPRAS is known, yet it is not known to be as hard as #SAT. The authors achieve this equivalence by constructing polynomial‑time AP‑reductions that translate the list‑colouring instance into a #BIS instance, exploiting the ordering or interval representations that characterize bipartite permutation and proper interval graphs.

3. #SAT‑equivalent case – For all other connected H, approximating #List‑H‑Col is AP‑equivalent to #SAT, the canonical #P‑complete counting problem. Hence, unless RP = NP, no FPRAS exists for these H. The proof proceeds by showing that any such H contains a hard induced subgraph (e.g., K′₂) that can be forced via appropriate colour lists, allowing a reduction from #SAT to #List‑H‑Col.

The authors also extend the trichotomy to bounded‑degree inputs (Theorem 2). For any maximum degree Δ ≥ 6, the same three regimes hold for #List‑H‑Col(Δ). Moreover, for reflexive or irreflexive H the hardness already appears for Δ ≥ 3. The Δ ≥ 6 condition is necessary because certain graphs (e.g., K′₂) admit an FPT‑approximation scheme when the degree bound is ≤ 5.

A notable aspect of the work is that the classification is expressed in terms of hereditary graph classes (bipartite permutation graphs and proper interval graphs), giving a natural graph‑theoretic formulation. Furthermore, the proofs are largely self‑contained and avoid heavy universal‑algebra machinery (such as multimorphisms) that appears in related weighted‑list colouring dichotomies. Instead, the authors rely on combinatorial reductions, the theory of hereditary classes, and known approximation results for the antiferromagnetic Ising model on bounded‑degree graphs.

The paper also discusses the relationship to the un‑list version #H‑Col. While exact counting complexities coincide for the list and non‑list versions, the approximation complexities diverge: many H for which #H‑Col is #BIS‑hard become #SAT‑hard when lists are allowed, as illustrated by the “2‑wrench” example. This underscores the added expressive power of lists in encoding hard substructures.

In summary, the authors provide a complete, elegant trichotomy for the approximate counting of list H‑colourings, identifying precisely when the problem is easy (polynomial time), of intermediate difficulty (#BIS‑equivalent), or as hard as any #P problem (#SAT‑equivalent). The results hold both in the unrestricted and bounded‑degree settings, and the techniques offer a blueprint for analyzing other list‑based counting problems without resorting to deep algebraic tools.