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：本文提出针对图着色问题（GCP）和带宽着色问题（BCP）中基于偏序的整数线性规划（ILP）模型的新型SAT编码。GCP要求为给定图的顶点分配最少颜色数，使得任意两个相邻顶点颜色不同；BCP是其推广形式，其中每条边具有权重，强制所分配颜色间保持最小"距离"，目标是使使用的"最大"颜色最小化。对于广泛研究的GCP，我们在DIMACS基准测试集上通过实验将新型SAT编码与最先进方法进行了比较。评估证实该SAT编码对稀疏图有效，甚至在某些DIMACS实例上优于现有最优方法。针对BCP，理论分析表明基于偏序的SAT与ILP公式的渐进规模小于经典分配模型。实际评估不仅证实了该编码相对于分配类编码的优势，还表明其在多个基准实例上优于当前最优方法。据我们所知，本文首次解决了文献中多个BCP公开实例。