In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.

翻译：本文针对图着色问题（GCP）和带宽着色问题（BCP），提出了基于偏序整数线性规划模型的新型SAT编码方法。图着色问题要求为给定图的顶点分配最少种类的颜色，使得任意两个相邻顶点获得不同颜色。带宽着色问题是其推广形式，其中每条边具有一个权重，强制要求所分配颜色之间的最小“距离”，目标是最小化所使用的“最大”颜色值。针对已被广泛研究的图着色问题，我们在DIMACS基准测试集上实验比较了新型SAT编码与现有最优方法。评估结果表明，该SAT编码对稀疏图具有良好效果，甚至在部分DIMACS实例上超越了现有最优方法。对于带宽着色问题，理论分析表明基于偏序的SAT和整数线性规划模型在渐进规模上小于经典的基于赋值的模型。实际评估不仅证实了相较于基于赋值编码的优势，还在基准实例集上超越了现有最优方法。据我们所知，本研究首次解决了文献中多个未解决的带宽着色问题实例。