Quasi-Stable Coloring for Graph Compression

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.

Blind Extraction of Equitable Partitions from Graph Signals

“…Finally, using a less algorithmic perspective, one can define the problem of approximating EP by specifying a tolerance ϵ of allowed deviation from eq. ( 1) and consequently asking for the minimimum number of clusters that still satisfy this constraint [20]. In this paper, we adopt the opposite approach and instead specify a number of clusters k and then ask for the partition minimizing a cost function (section 4) i.e.…”