Parameterized and Approximation Algorithms for the Load Coloring Problem

TL;DR

This paper extends the study of the Load Coloring Problem to any fixed number of colors, providing linear kernels and approximation algorithms, thereby establishing fixed parameter tractability for all fixed c.

Contribution

It introduces linear kernels for the $(c,k)$-Load Coloring Problem for any fixed c, generalizing previous results for c=2, and develops approximation algorithms with constant ratios.

Findings

Linear-vertex and linear-edge kernels for $(c,k)$-LCP for fixed c

$(c,k)$-LCP is fixed parameter tractable for all fixed c

Approximation algorithms with constant ratios for the optimization version

Abstract

Let $c, k$ be two positive integers and let $G=(V,E)$ be a graph. The $(c,k)$-Load Coloring Problem (denoted $(c,k)$-LCP) asks whether there is a $c$-coloring $\varphi: V \rightarrow [c]$ such that for every $i \in [c]$, there are at least $k$ edges with both endvertices colored $i$. Gutin and Jones (IPL 2014) studied this problem with $c=2$. They showed $(2,k)$-LCP to be fixed parameter tractable (FPT) with parameter $k$ by obtaining a kernel with at most $7k$ vertices. In this paper, we extend the study to any fixed $c$ by giving both a linear-vertex and a linear-edge kernel. In the particular case of $c=2$, we obtain a kernel with less than $4k$ vertices and less than $8k$ edges. These results imply that for any fixed $c\ge 2$, $(c,k)$-LCP is FPT and that the optimization version of $(c,k)$-LCP (where $k$ is to be maximized) has an approximation algorithm with a constant ratio for any fixed $c\ge 2$.