Abstract.

The approximate graph coloring problem, whose complexity is unresolved in most cases, concerns finding a \(c\) -coloring of a graph that is promised to be \(k\) -colorable, where \(c\geq k\) . This problem naturally generalizes to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyze the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph coloring and promise graph homomorphism problems.

Keywords

  1. graph coloring
  2. graph homomorphism problem
  3. constraint satisfaction
  4. polymorphism
  5. promise problem

MSC codes

  1. 68Q17
  2. 68Q25
  3. 68R05
  4. 05C15

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Acknowledgments.

A. K. and J. O. would like to thank John Hunton for consultations on algebraic topology and Libor Barto and Antoine Mottet for inspiring discussions.

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Information & Authors

Information

Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 38 - 79
ISSN (online): 1095-7111

History

Submitted: 3 November 2020
Accepted: 23 September 2022
Published online: 14 February 2023

Keywords

  1. graph coloring
  2. graph homomorphism problem
  3. constraint satisfaction
  4. polymorphism
  5. promise problem

MSC codes

  1. 68Q17
  2. 68Q25
  3. 68R05
  4. 05C15

Authors

Affiliations

Department of Computer Science, Durham University, UK.
Institute of Science and Technology Austria, Klosterneuburg, Austria.
Faculty of Mathematics, Informatics, and Mechanics, Unversity of Warsaw, Poland.
Department of Computer Science, University of Oxford, Oxford, UK.

Funding Information

ERC
Funding: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 714532). The first and second authors were supported by the UK EPSRC grant EP/R034516/1. The fourth author was supported by a Royal Society University Research Fellowship. The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.

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