Hardness of the Generalized Coloring Numbers

Abstract

The generalized coloring numbers of Kierstead and Yang [7] offer an algorithmically useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we show that it is NP-hard to compute the weak 2-coloring number (answering an open question of Grohe et al. [5]). We then complete the picture by proving that the r-coloring number is also NP-hard to compute for all r ≥ 2. Finally, we give an approximation algorithm for the r-coloring number which improves both the runtime and approximation factor of the existing approach of Dvořák [3]. Our algorithm greedily orders vertices with small enough i-reach for every i ≤ r and achieves an O(C r−1 k r−1 )-approximation, where C j is the jth Catalan number.

Turbocharging Heuristics for Weak Coloring Numbers

“…The limitation of degeneracy is, however, that as soon as we want to use it for less local problems than enumerating cliques it is unlikely that we can give efficient running-time guarantees [KS15]. This problem is alleviated by weak r-coloring numbers [RS20], but for each r ≥ 2 computing the weak r-coloring number is NP-complete [Gro+18;BLS21].…”