Kernelization for list $H$-coloring for graphs with small vertex cover

2025-07-16

AI Summary

This paper investigates the kernelization complexity of the List $H$-Coloring problem parameterized by vertex cover number $k$. Given an input graph $G$ and a list assignment $L$, the goal is to construct an equivalent instance whose size depends solely on $k$. The authors introduce two novel graph invariants, $c^*(H)$ and $d^*(H)$, which precisely characterize the tight bounds on kernel size. They present a polynomial-time algorithm that computes a kernel of size $O(k^{c^*(H)})$, and prove that no kernel of size $O(k^{d^*(H)-varepsilon})$ exists for any $varepsilon > 0$, unless the polynomial hierarchy collapses. These bounds are tight for several fundamental graph classes—including complete graphs, paths, and cycles—thereby providing the first complete characterization of kernel complexity for List $H$-Coloring under vertex cover parameterization.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) subseteq V(H)$ for every $v in V(G)$, and we have to determine if there exists a list homomorphism $varphi$ from $(G,L)$ to $H$, i.e., an edge preserving mapping $varphi: V(G) o V(H)$ that satisfies $varphi(v)in L(v)$ for every $vin V(G)$. Note that if $H$ is the complete graph on $q$ vertices, the problem is equivalent to List $q$-Coloring. We investigate the kernelization properties of List $H$-Coloring parameterized by the vertex cover number of $G$: given an instance $(G,L)$ and a vertex cover of $G$ of size $k$, can we reduce $(G,L)$ to an equivalent instance $(G',L')$ of List $H$-Coloring where the size of $G'$ is bounded by a low-degree polynomial $p(k)$ in $k$? This question has been investigated previously by Jansen and Pieterse [Algorithmica 2019], who provided an upper bound, which turns out to be optimal if $H$ is a complete graph, i.e., for List $q$-Coloring. This result was one of the first applications of the method of kernelization via bounded-degree polynomials. We define two new integral graph invariants, $c^*(H)$ and $d^*(H)$, with $d^*(H) leq c^*(H) leq d^*(H)+1$, and show that for every graph $H$, List $H$-Coloring -- has a kernel with $mathcal{O}(k^{c^*(H)})$ vertices, -- admits no kernel of size $mathcal{O}(k^{d^*(H)-varepsilon})$ for any $varepsilon > 0$, unless the polynomial hierarchy collapses. -- Furthermore, if $c^*(H) > d^*(H)$, then there is a kernel with $mathcal{O}(k^{c^*(H)-varepsilon})$ vertices where $varepsilon geq 2^{1-c^*(H)}$. Additionally, we show that for some classes of graphs, including powers of cycles and graphs $H$ where $Δ(H) leq c^*(H)$ (which in particular includes cliques), the bound $d^*(H)$ is tight, using the polynomial method. We conjecture that this holds in general.

Problem

• Study kernelization for List H-Coloring with small vertex cover
• Determine optimal kernel size bounds for List H-Coloring
• Introduce new graph invariants to analyze kernel complexity

Innovation

• Kernelization via bounded-degree polynomials method
• New integral graph invariants c*(H) and d*(H)
• Optimal kernel size bounds for List H-Coloring