Sparsification Lower Bounds for List-Coloring

Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Pawel Rzazewski

Abstract: We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize \(\mathcal {O} (n^{2-\varepsilon })\) (n2-ɛ) for some ɛ > 0? We prove that if  H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless \(\mathsf {NP \subseteq coNP/poly}\). Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-graphs.