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.

我们引入了一个迭代框架，用于使用决策图解决图形着色问题。决策图紧凑地表示所有可能的颜色类别，其中一些可能包含边缘冲突。在每次迭代中，我们使用约束最小网络流量模型来计算下限并确定冲突。通过完善决策图，可以消除与这些冲突相关的不可行的颜色类别。我们证明，在最佳情况下，我们的方法可能会使用比精确图小的指数图来证明最优性。我们还基于决策图开发原始启发式算法，以在每次迭代时找到可行的解决方案。我们对所有137个DIMACS图形着色实例进行了实验评估。我们的程序可以最佳地解决52个实例，其中44个实例在1分钟内解决。我们还将我们的方法与基于分支和价格的最新图形着色求解器进行了比较，并表明我们获得了竞争性结果。最后，我们报告了开放实例的下限有所改善C2000.9。