Sparsification Lower Bounds for List $H$-Coloring

TL;DR

This paper establishes lower bounds for sparsification of the List $H$-Coloring problem, showing that non-bi-arc graphs cannot be efficiently sparsified unless unlikely complexity collapses occur.

Contribution

It proves that List $H$-Coloring for non-bi-arc graphs does not admit polynomial sparsification unless NP is in coNP/poly, combining kernelization lower bounds with graph structure analysis.

Findings

No polynomial sparsification for non-bi-arc graphs unless NP ⊆ coNP/poly.

Uses kernelization lower bounds and graph structure analysis.

Extends understanding of complexity boundaries for List $H$-Coloring.

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 $v \in V(G)$ is mapped to a vertex on its list $L(v) \subseteq V(H)$. 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 $O(n^{2-\varepsilon})$ for some $\varepsilon > 0$? We prove that if $H$ is not a bi-arc graph, then List $H$-Coloring does not admit such a sparsification algorithm unless $NP \subseteq coNP/poly$. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs $H$ which are not bi-arc graphs.