Turbocharging Heuristics for Weak Coloring Numbers

TL;DR

This paper explores advanced algorithms to improve the computation of weak coloring numbers in graphs, enhancing heuristic methods with exponential-time procedures, leading to better results on real-world graph benchmarks.

Contribution

It introduces a novel approach combining heuristics with exponential-time subprocedures for computing weak coloring numbers, with proven hardness and tractability results.

Findings

Improved weak coloring numbers on over half of benchmark instances.

Algorithms are competitive with previous methods on large graphs.

Enhanced heuristics yield better bounds in practical scenarios.

Abstract

Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number is breached. We provide hardness and tractability results on the corresponding computational subproblems. We implemented several of the resulting algorithms and show them to be competitive with previous approaches on a previously studied set of benchmark instances containing 86 graphs with up to 183831 edges. We obtain improved weak coloring numbers for over half of the instances.