We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work-efficiency. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring.

中文翻译：

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对工作、深度和质量有强大保证的高性能并行图着色

我们开发了第一个平行图着色启发式算法，在工作、深度和着色质量方面具有强大的理论保证。关键思想是设计顶点退化顺序的松弛，这是一个众所周知的图论概念，并按照这种松弛所指示的顺序为顶点着色。这将可调整数量的并行性引入到难以并行化的退化排序中。这个简单的想法在图形着色的几个关键方面带来了显着的好处。例如，我们的一种算法可确保多对数深度和所用颜色数量的界限，这优于所有其他可并行化方案，同时保持工作效率。除了可证明的保证外，开发的算法对几个真实世界的图形具有竞争性的运行时间，同时几乎总是提供卓越的着色质量。我们的简并排序松弛对于着色上下文之外的算法具有单独的兴趣。