Color refinement is a crucial subroutine in symmetry detection in theory as well as practice. It has further applications in machine learning and in computational problems from linear algebra. While tight lower bounds for the worst case complexity are known [Berkholz, Bonsma, Grohe, ESA2013] no comparative analysis of design choices for color refinement algorithms is available. We devise two models within which we can compare color refinement algorithms using formal methods, an online model and an approximation model. We use these to show that no online algorithm is competitive beyond a logarithmic factor and no algorithm can approximate the optimal color refinement splitting scheme beyond a logarithmic factor. We also directly compare strategies used in practice showing that, on some graphs, queue based strategies outperform stack based ones by a logarithmic factor and vice versa. Similar results hold for strategies based on priority queues.

中文翻译：

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最坏情况下颜色优化算法的比较设计选择分析

在理论和实践中，颜色优化是对称检测中至关重要的子程序。它在机器学习和线性代数的计算问题中有进一步的应用。虽然已知最坏情况复杂性的严格下限[Berkholz，Bonsma，Grohe，ESA2013]，但无法进行颜色优化算法的设计选择比较分析。我们设计了两个模型，可以使用形式化方法比较颜色细化算法，一个是在线模型，一个是近似模型。我们使用这些数据来表明，没有在线算法具有超越对数因子的竞争能力，并且没有任何算法能够超过具有对数因子的近似最佳色彩细化拆分方案。我们还直接比较了实践中使用的策略，这些策略表明，在某些图表上，基于队列的策略的对数因数优于基于堆栈的策略，反之亦然。基于优先级队列的策略也有类似的结果。