Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case

TL;DR

This paper compares different design choices for color refinement algorithms using formal models, revealing limitations of online algorithms and highlighting when certain practical strategies outperform others.

Contribution

It introduces formal models for comparing color refinement algorithms and provides theoretical bounds and practical insights into their performance.

Findings

No online algorithm is competitive beyond a logarithmic factor.

Algorithms cannot approximate optimal splitting beyond a logarithmic factor.

Queue-based strategies can outperform stack-based ones on some graphs.

Abstract

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.