Hardness of the generalized coloring numbers

Michael Breen-McKay, Brian Lavallee, Blair D. Sullivan

Abstract: The generalized coloring numbers of Kierstead and Yang (Order 2003) 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 complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak r<math><mi is="true">r</mi></math>-coloring number at most k<math><mi is="true">k</mi></math> is para-NP-hard when parameterized by k<math><mi is="true">k</mi></math> for all r≥2<math><mrow is="true"><mi is="true">r</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">2</mn></mrow></math>. We adapt this to determining if a graph has r<math><mi is="true">r</mi></math>-coloring number at most k<math><mi is="true">k</mi></math> as well, proving para-NP-hardness for all r≥2<math><mrow is="true"><mi is="true">r</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">2</mn></mrow></math>. Para-NP-hardness implies that no XP algorithm (runtime O(nf(k))<math><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">f</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></msup><mo is="true">)</mo></mrow></mrow></math>) exists for testing if a generalized coloring number is at most k<math><mi is="true">k</mi></math>. Moreover, there exists a constant c<math><mi is="true">c</mi></math> such that it is NP-hard to approximate the generalized coloring numbers within a factor of c<math><mi is="true">c</mi></math>. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvořák (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a (k−1)r−1<math><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow><mrow is="true"><mi is="true">r</mi><mo is="true">−</mo><mn is="true">1</mn></mrow></msup></math>-approximation for the r<math><mi is="true">r</mi></math>-coloring number and an O(kr−1)<math><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mi is="true">r</mi><mo is="true">−</mo><mn is="true">1</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math>-approximation for the weak r<math><mi is="true">r</mi></math>-coloring number.