超越最坏情况的颜色细化算法的比较设计选择分析
Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case
摘要：
颜色细化是对称性检测中的一个重要子程序，无论在理论上还是在实践中都是如此.它在机器学习和线性代数的计算问题中有进一步的应用。虽然已知最差情况复杂度的严格下限[Berkholz，Bonsma，Grohe，ESA 2013]，但没有对颜色细化算法的设计选择进行比较分析。我们设计了两个模型，我们可以比较颜色细化算法使用正式的方法，在线模型和近似模型。我们使用这些来表明，没有在线算法是有竞争力的对数因子之外，没有算法可以近似的最佳颜色细化分裂方案超过对数因子。我们还直接比较在实践中使用的策略，在某些图形上，基于队列的策略优于基于堆栈的对数因子，反之亦然。类似的结果也适用于基于优先级队列的策略。
英文摘要：
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.
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