Turbocharging Heuristics for Weak Coloring Numbers

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 55 graphs with up to 8581 edges. We obtain improved weak coloring numbers for half of the instances. We also establish a baseline for lower bounds on the coloring numbers. * This paper is based on Alexander Dobler's master thesis [Dob21].