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Sparsification Lower Bounds for List H-Coloring

Authors Hubie Chen, Bart M. P. Jansen , Karolina Okrasa , Astrid Pieterse , Paweł Rzążewski

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Author Details

Hubie Chen
  • Birkbeck, University of London, Malet Street, Bloomsbury, UK
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Karolina Okrasa
  • University of Warsaw, Institute of Informatics, Poland
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
Astrid Pieterse
  • Department of Computer Science, Humboldt-Universität zu Berlin, Germany
Paweł Rzążewski
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
  • University of Warsaw, Institute of Informatics, Poland

Funding

  • Jansen, Bart M. P.: Supported by NWO Gravitation grant "Networks".
  • Okrasa, Karolina: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement no. 714704.
  • Pieterse, Astrid: Supported by DFG Emmy Noether-grant (KR 4286/1), part of this research was supported by NWO Gravitation grant "Networks".
  • Rzążewski, Paweł: Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

Acknowledgements

We acknowledge the productive atmosphere at Dagstuhl Seminar 19271 "Graph Colouring: from Structure to Algorithms", where this work has been initiated.

Cite AsGet BibTex

Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, and Paweł Rzążewski. Sparsification Lower Bounds for List H-Coloring. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 58:1-58:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.58

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 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • List H-Coloring
  • Sparsification
  • Constraint Satisfaction Problem

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