Bounds and Fixed-Parameter Algorithms for Weighted Improper Coloring (Extended Version)

TL;DR

This paper explores the computational complexity of weighted improper coloring, establishing hardness results and providing fixed-parameter algorithms for specific graph parameters, including a linear-time solution for interval graphs with bounded degree.

Contribution

It introduces new fixed-parameter algorithms for weighted improper coloring based on treewidth and maximum degree, and extends bounds from defective coloring to the weighted case.

Findings

Weighted improper coloring is not fixed-parameter tractable by pathwidth.

Provides fixed-parameter algorithms parameterized by treewidth and maximum degree.

Offers a linear-time algorithm for interval graphs with bounded degree.

Abstract

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge weights. Finally we give fixed-parameter algorithms for weighted improper coloring both when parameterized by treewidth and maximum degree and when parameterized by treewidth and precision of edge weights. In particular, we obtain a linear-time algorithm for weighted improper coloring of interval graphs of bounded degree.