一种颜色预处理改进DSATUR

中文摘要 AI 辅助

图着色问题（GCP）是NP难的，DSATUR是解决该问题最快的启发式算法之一，尽管其产生的着色通常比最先进的着色算法使用更多颜色。我们提出SSLD（基于DSATUR的半定谱学习），该方法通过预处理第一个好的颜色类，然后让DSATUR完成对给定图其余部分的着色，从而改进DSATUR。我们从半定规划（SDP）中获得该颜色类，类似于用于计算Lovász theta数的SDP。据我们所知，SSLD是第一种通过固定颜色类预处理来改进DSATUR的方法。我们在DIMACS实例、随机图（Erdős–Rényi、Watts-Strogatz、Barabási–Albert）、频率分配和作业车间调度实例上，将SSLD与DSATUR以及一种朴素的1颜色类预处理算法进行评估。在超过1600个基准实例中，SSLD在几乎所有情况下都匹配或击败DSATUR，并且优于朴素的GISD基线，这使我们能够确认SDP引导选择第一个颜色类所带来的价值。这种质量以比DSATUR慢约195倍的运行时间为代价，但证明了SDP引导的预处理第一个颜色类是未来改进的一个方向。

英文摘要

The Graph Coloring Problem (GCP) is NP-hard and DSATUR stands as one of the fastest heuristics for it despite producing colorings that typically use more colors than state-of-the-art coloring algorithms. We propose SSLD (Semidefinite Spectral Learning with DSATUR), which improves DSATUR by preprocessing a first good color class before letting DSATUR complete coloring the rest of the given graph. We obtain this color class from a Semidefinite Programming (SDP), similar to an SDP used to compute the Lovász theta number. To the best of our knowledge, SSLD is the first approach to improve DSATUR by preprocessing through fixed color classes. We evaluate SSLD against DSATUR and against a naive 1-color-class preprocessing algorithm on DIMACS instances, random graphs (Erdős--Rényi, Watts-Strogatz, Barabási--Albert), Frequency Assignment and Job Shop Scheduling instances. SSLD matches or beats DSATUR in almost every case across over 1600 benchmark instances, and out performs the naive GISD baseline, allows us to confirm the value brought by the SDP-guided choice of the first color class. This quality comes at a runtime cost of roughly 195 times slower that DSATUR, but demonstrating that SDP-guided preprocessing of a first color class is a direction for future improvements.

发表机构

• Université de Strasbourg（斯特拉斯堡大学）

机构由 AI 辅助整理，请以论文原文为准。