Full complexity classification of the list homomorphism problem for bounded-treewidth graphs
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A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ∈ V(G) it holds that h(v) ∈ L(v). The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. We are interested in the complexity of the problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rzążewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H. In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHomH problem is NP-hard. For every such H we find a constant k = k(H), such that LHom(H) on instances with n vertices and treewidth t * can be solved in time k^t· n^𝒪(1), provided that the input graph is given along with a tree decomposition of width t, * cannot be solved in time (k-ε)^t· n^𝒪(1), for any ε >0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|. Obtaining matching upper and lower bounds shows that the set of algorithmic tools we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of LHom(H), e.g. with different parameterizations.
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