Parameterized and Approximation Algorithms for the Load Coloring Problem

Florian Barbero, Gregory Z. Gutin, Mark Jones, Bin Sheng

Abstract: Let c, k be two positive integers. Given a graph \(G=(V,E)\), the c-Load Coloring problem 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 (Inf Process Lett 114:446–449, 2014) studied this problem with \(c=2\). They showed 2-Load Coloring 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 \(6k+(3+\sqrt{2})\sqrt{k}+4\)  edges. These results imply that for any fixed \(c\ge 2\), c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.