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Optimality Clue for Graph Coloring Problem

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Integration of Constraint Programming, Artificial Intelligence, and Operations Research (CPAIOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11494))

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Abstract

In this paper, a new approach is presented to qualify or not, a solution found by a heuristic for a potential optimal solution. Our approach is based on the following observation: for a minimization problem, the number of admissible solutions decreases with the value of the objective function. Concerning the Graph Coloring Problem (GCP), we confirm this observation and present a new way in which to prove optimality. This proof is based on the counting of the number of different k-colorings and the number of independent sets of a given graph G.

Finding the exact solution for counting problems is difficult (#P-complete). However, we show that in using only randomized heuristics, it is possible to define an estimation of the upper bound of the number of k-colorings. This estimate has been calibrated on a broad benchmark of graph instances for which the exact number of optimal k-colorings is known.

Our approach, called optimality clue, constructs a sample of k-colorings from a given graph by running one randomized heuristic a number of times on the same graph instance. We use the evolutionary algorithm HEAD [26], which is one of the most efficient heuristics for GCP.

Optimality clue matches the standard definition of optimality on a wide number of instances for DIMACS and RBCII benchmarks where the optimality is known.

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Notes

  1. 1.

    Notice that it is still NP-hard to approximate \(\chi (G)\) within \(n^{1-\epsilon }\) for any \(\epsilon > 0\) [35].

  2. 2.

    A perfect graph is a graph for which the chromatic number of every induced subgraph is the same as the size of the largest clique of that subgraph. 1-perfect graphs are more general than perfect graphs.Polynomial-time exact algorithms with the aim of finding \(\chi (G)\) for perfect graphs [13] exist, but are in reality slow in performance. Line graphs, chordal graphs, interval graphs or cographs are subclasses of perfect graphs.

  3. 3.

    An independent set is a subset of vertices of G, so every two distinct vertices in the independent set are not adjacent.

  4. 4.

    Open-source code available at: github.com/graphcoloring/HEAD.

  5. 5.

    A maximal clique is a clique that cannot be extended by including an additional adjacent vertex. A maximum clique is a clique that has the largest size in a given graph; a maximum clique is therefore always maximal, but the converse is not so. Analogue definition for IS.

  6. 6.

    Code available at: https://users.aalto.fi/~pat/cliquer.html. To count all IS of a graph, you just execute: ./cl \({<}\)complement graph\({>}\) -a -m 1 -M \({<}k{>}\).

  7. 7.

    All k-colorings in the sample are uniformly drawn at random in \(\varOmega (G,k)\).

  8. 8.

    The fitness landscape itself depends on the neighborhood used for both the tabu search and the crossover.

  9. 9.

    This problem is linked to the birthday problem that shows that in a room of just 23 people there’s a 50-50 chance that two people have the same birthday. In our case, the number of days in a year is \(\mathcal {N}\) and the number of people is the size t of the sample.

  10. 10.

    Code available at: lamsade.dauphine.fr/coloring/doku.php.

  11. 11.

    Instances available in the same address.

  12. 12.

    For <DSJC500.5> the computation time is not reported because it takes several weeks and no accurate time has been recorded.

  13. 13.

    In this context, we propose on our website a challenge to find a counterexample (false positive graph).

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Authors and Affiliations

  1. ENAC, French Civil Aviation University, Toulouse, France

    Alexandre Gondran

  2. UHA, University of Haute Alsace, Mulhouse, France

    Laurent Moalic

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  1. Ecole Polytechnique de Montréal, Montréal, QC, Canada

    Louis-Martin Rousseau

  2. University of Western Macedonia, Kozani, Greece

    Kostas Stergiou

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Gondran, A., Moalic, L. (2019). Optimality Clue for Graph Coloring Problem. In: Rousseau, LM., Stergiou, K. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019. Lecture Notes in Computer Science(), vol 11494. Springer, Cham. https://doi.org/10.1007/978-3-030-19212-9_22

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