Quasi-stable Coloring for Graph Compression: Approximating Max-Flow, Linear Programs, and Centrality

TL;DR

This paper introduces quasi-stable coloring, an approximate graph coloring method that improves graph compression and approximation for linear programming, max-flow, and centrality measures, balancing accuracy and compression.

Contribution

It presents the first approximate color refinement scheme, quasi-stable coloring, enabling better graph compression and approximation in applications like LP, max-flow, and centrality.

Findings

Quasi-stable coloring provides good approximations on reduced graphs.

The problem of computing maximal quasi-stable coloring is NP-hard.

Experimental results show effectiveness compared to prior techniques.

Abstract

We propose quasi-stable coloring, an approximate version of stable coloring. Stable coloring, also called color refinement, is a well-studied technique in graph theory for classifying vertices, which can be used to build compact, lossless representations of graphs. However, its usefulness is limited due to its reliance on strict symmetries. Real data compresses very poorly using color refinement. We propose the first, to our knowledge, approximate color refinement scheme, which we call quasi-stable coloring. By using approximation, we alleviate the need for strict symmetry, and allow for a tradeoff between the degree of compression and the accuracy of the representation. We study three applications: Linear Programming, Max-Flow, and Betweenness Centrality, and provide theoretical evidence in each case that a quasi-stable coloring can lead to good approximations on the reduced graph. Next, we consider how to compute a maximal quasi-stable coloring: we prove that, in general, this problem is NP-hard, and propose a simple, yet effective algorithm based on heuristics. Finally, we evaluate experimentally the quasi-stable coloring technique on several real graphs and applications, comparing with prior approximation techniques. A reference implementation and the experiment code are available at https://github.com/mkyl/QuasiStableColors.jl .