In the ``list homomorphism'' problem, the input consists of two graphs G and H, together with a list L(v) of vertices of H for every vertex v in V(G). The task is to find a homomorphism phi:V(G)->V(H) respecting the lists, that is, we have that phi(v) is in L(v) for every v in V(H), and if u and v are adjacent in G, then phi(u) and phi(v) are adjacent in H. If H is a fixed graph, then the problem is denoted LHOM(H).

We consider the ``reflexive'' version of the problem, where we assume that every vertex in H has a self-loop. It is known that reflexive LHOM(H) is polynomial-time solvable if H is an interval graph and it is NP-complete otherwise [Feder and Hell, JCTB 1998].

We explore the complexity of the problem parameterized by the treewidth tw(G) of the input graph G. If a tree decomposition of G of width tw(G) is given in the input, then the problem can be solved in time |V(H)|^{tw(G)}n^{O(1)} by naive dynamic programming. Our main result completely reveals when and by exactly how much this naive algorithm can be improved. We introduce a simple combinatorial invariant i*(H), which is based on the existence of decompositions and incomparable sets, and show that this number should appear as the base of the exponent in the best possible running time. Specifically, we prove for every fixed graph H that

--- If a tree decomposition of width tw(G) is given in the input, then the problem can be solved in time i*(H)^{tw(G)}n^{O(1)}.

--- Assuming the Strong Exponential-Time Hypothesis (SETH), the problem cannot be solved in time (i*(H)-c)^{tw(G)}n^{O(1)} for any c>0.

Thus by matching upper and lower bounds, our result exactly characterizes for every fixed H the complexity of reflexive LHOM(H) parameterized by treewidth.

These results are joint work with Laszlo Ergi and Daniel Marx.