List Homomorphism: Beyond the Known Boundaries

Sriram Bhyravarapu, Satyabrata Jana, Fahad Panolan, Saket Saurabh, Shaily Verma

Abstract: Given two graphs G and H, and a list \(L(u)\subseteq V(H)\) associated with each \(u\in V(G)\), a list homomorphism from G to H is a mapping \(f:V(G)\rightarrow V(H)\) such that (i) for all \(u\in V(G)\), \(f(u) \in L(u)\), and (ii) for all \(u,v\in V(G)\), if \(uv\in E(G)\) then \(f(u)f(v)\in E(H)\). The List Homomorphism  problem asks whether there exists a list homomorphism from G to H. Enright, Stewart and Tardos [SIAM J. Discret. Math., 2014] showed that the List Homomorphism  problem can be solved in \(O(n^{k^2-3k+4})\) time on graphs where every connected induced subgraph of G admits “a multichain ordering” (see the introduction for the definition of multichain ordering of a graph), that includes permutation graphs, biconvex graphs, and interval graphs, where \(n=|V(G)|\) and \(k=|V(H)|\). We prove that List Homomorphism parameterized by k even when G is a bipartite permutation graph is W[1]-hard. In fact, our reduction implies that it is not solvable in time \(n^{o(k)}\), unless the Exponential Time Hypothesis (ETH) fails. We complement this result with a matching upper bound and another positive result.