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We present an exact algorithm for graph coloring and maximum clique problems based on SAT technology. It relies on four sub-algorithms that alternatingly compute cliques of larger size and colorings with fewer colors. We show how these techniques can mutually help each other: larger cliques facilitate finding smaller colorings, which in turn can boost finding larger cliques. We evaluate our approach on the DIMACS graph coloring suite. For finding maximum cliques, we show that our algorithm can improve the state-of-the-art MaxSAT-based solver IncMaxCLQ, and for the graph coloring problem, we close two open instances, decrease two upper bounds, and increase one lower bound.
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Graph coloring with decision diagrams
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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- Award ID(s):
- 1918102
- PAR ID:
- 10292261
- Date Published:
- Journal Name:
- Mathematical programming
- ISSN:
- 1436-4646
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10107-021-01662-x
