Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Paweł Rzążewski
{"title":"Sparsification Lower Bounds for List <i>H</i> -Coloring","authors":"Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Paweł Rzążewski","doi":"10.1145/3612938","DOIUrl":null,"url":null,"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 }) \\) for some ε > 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.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Computation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3612938","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 1
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 \(\mathcal {O}(n^{2-\varepsilon }) \) for some ε > 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.