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Abstract. We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

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In the deletion version of the list homomorphism problem, we are given graphs G and H, a list L(v) ⊆ V (H) for each vertex v ∈ V (G), and an integer k. The task is to decide whether there exists a set W ⊆ V (G) of size at most k such that there is a homomorphism from G \ W to H respecting the lists. We show that DL-Hom(H), parameterized by k and |H|, is fixed-parameter tractable for any (P6, C6)-free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom(H) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. [9], a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NPcomplete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N , and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is hard for NL. Our algorithm operates by reducing the lists in a complex manner based on a novel decomposition of an auxiliary digraph, combined with repeated applications of Reingold's algorithm for undirected reachability (2005). We also prove an algebraic version of this dichotomy: the digraphs without a circular N are precisely those that admit a finite chain of conservative polymorphisms satisfying the Hagemann-Mitschke identities. This confirms a conjecture of Larose and Tesson (2007) for LHOM (H). Moreover, we show that the presence of a circular N can be decided in time polynomial in the size of H.