Hardness of the Generalized Coloring Numbers

TL;DR

This paper investigates the computational hardness of generalized coloring numbers, proving NP-hardness and para-NP-hardness results, and introduces improved approximation algorithms for these graph parameters.

Contribution

It establishes the NP-hardness and para-NP-hardness of computing generalized coloring numbers and provides improved approximation algorithms with better runtime and approximation guarantees.

Findings

Computing weak 2-coloring number is NP-hard.

Determining if a graph has weak r-coloring number at most k is para-NP-hard for all r ≥ 2.

A greedy ordering algorithm achieves a (k-1)^{r-1}-approximation for r-coloring number.

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 $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as well, proving para-NP-hardness for all $r \geq 2$. Para-NP-hardness implies that no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if a generalized coloring number is at most $k$. Moreover, there exists a constant $c$ such that it is NP-hard to approximate the generalized coloring numbers within a factor of $c$. 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 Dvo\v{r}\'{a}k (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a $(k-1)^{r-1}$-approximation for the $r$-coloring number and an $O(k^{r-1})$-approximation for the weak $r$-coloring number.