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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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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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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.

Cite As Get 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

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.

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