eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
58:1
58:17
10.4230/LIPIcs.ISAAC.2020.58
article
Sparsification Lower Bounds for List H-Coloring
Chen, Hubie
1
Jansen, Bart M. P.
2
https://orcid.org/0000-0001-8204-1268
Okrasa, Karolina
3
4
https://orcid.org/0000-0003-1414-3507
Pieterse, Astrid
5
https://orcid.org/0000-0003-3721-6721
Rzążewski, Paweł
4
3
https://orcid.org/0000-0001-7696-3848
Birkbeck, University of London, Malet Street, Bloomsbury, UK
Eindhoven University of Technology, The Netherlands
University of Warsaw, Institute of Informatics, Poland
Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
Department of Computer Science, Humboldt-Universität zu Berlin, Germany
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 𝒪(n^(2-ε)) 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 NP ⊆ 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.58/LIPIcs.ISAAC.2020.58.pdf
List H-Coloring
Sparsification
Constraint Satisfaction Problem