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Graph coloring with decision diagrams

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Abstract

We introduce an iterative framework for solving graph coloring problems using decision diagrams. The decision diagram compactly represents all possible color classes, some of which may contain edge conflicts. In each iteration, we use a constrained minimum network flow model to compute a lower bound and identify conflicts. Infeasible color classes associated with these conflicts are removed by refining the decision diagram. We prove that in the best case, our approach may use exponentially smaller diagrams than exact diagrams for proving optimality. We also develop a primal heuristic based on the decision diagram to find a feasible solution at each iteration. We provide an experimental evaluation on all 137 DIMACS graph coloring instances. Our procedure can solve 52 instances optimally, of which 44 are solved within 1 minute. We also compare our method to a state-of-the-art graph coloring solver based on branch-and-price, and show that we obtain competitive results. Lastly, we report an improved lower bound for the open instance C2000.9.

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Notes

  1. The operator \(+\) indicates the addition of a set to a set (instead of the union).

  2. The code has been downloaded from https://github.com/heldstephan/exactcolors.

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Acknowledgements

This work was partially supported by Office of Naval Research Grant No. N00014-18-1-2129 and National Science Foundation Award #1918102.

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  1. Carnegie Mellon University, Tepper School of Business, 5000 Forbes Avenue, Pittsburgh, PA, 15232, USA

    Willem-Jan van Hoeve

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van Hoeve, WJ. Graph coloring with decision diagrams. Math. Program. 192, 631–674 (2022). https://doi.org/10.1007/s10107-021-01662-x

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