Graph Coloring Lower Bounds from Decision Diagrams

Abstract

We introduce an iterative framework for computing lower bounds to graph coloring problems. We utilize relaxed decision diagrams to compactly represent an exponential set of color classes, or independent sets, some of which may contain edge conflicts. Our procedure uses minimum network flow models to compute lower bounds on the coloring number 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 provide an experimental evaluation on benchmark instances, and report an improved lower bound for one open instance.