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An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm

Abstract

We study the behavior of the Douglas–Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the Douglas–Rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the Douglas–Rachford algorithm, with this new approach, is demonstrated through various numerical experiments.

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Notes

  1. top95: http://magictour.free.fr/top95.

  2. DIMACS benchmark instances: http://cse.unl.edu/~tnguyen/npbenchmarks/graphcoloring.html.

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Acknowledgements

This work was initiated during the BIRS workshop on Splitting Algorithms, Modern Operator Theory, and Applications, organized by Heinz Bauschke, Regina Burachik and Russell Luke in Oaxaca (Mexico), in 2017. The authors thank the organizers for an excellent meeting and bringing us together.

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Correspondence to Francisco J. Aragón Artacho.

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F. J. Aragón Artacho and R. Campoy were partially supported by MICINN of Spain and ERDF of EU, Grants MTM2014-59179-C2-1-P and PGC2018-097960-B-C22. F. J. Aragón Artacho was supported by the Ramón y Cajal program by MINECO of Spain and ERDF of EU (RYC-2013-13327) and R. Campoy was supported by MINECO of Spain and ESF of EU (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”.

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Aragón Artacho, F.J., Campoy, R. & Elser, V. An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm. J Glob Optim 77, 383–403 (2020). https://doi.org/10.1007/s10898-019-00867-x

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  • DOI: https://doi.org/10.1007/s10898-019-00867-x

Keywords

  • Douglas–Rachford algorithm
  • Graph coloring
  • Feasibility problem
  • Nonconvex constraints

Mathematics Subject Classification

  • 47J25
  • 90C27
  • 47N10