The Complexity of the Partition Coloring Problem

TL;DR

This paper investigates the computational complexity of the Partition Coloring Problem, a generalization of vertex coloring, analyzing its difficulty based on parameters like colors, parts, and part size, and introduces a new exact algorithm.

Contribution

The paper determines the complexity status of the Partition Coloring Problem for various parameters and presents a novel exact algorithm for solving it.

Findings

Complexity results for different parameter combinations.

Identification of tractable and intractable cases.

A new exact algorithm for the problem.

Abstract

Given a simple undirected graph $G=(V,E)$ and a partition of the vertex set $V$ into $p$ parts, the \textsc{Partition Coloring Problem} asks if we can select one vertex from each part of the partition such that the chromatic number of the subgraph induced on the $p$ selected vertices is bounded by $k$. PCP is a generalized problem of the classical \textsc{Vertex Coloring Problem} and has applications in many areas, such as scheduling and encoding etc. In this paper, we show the complexity status of the \textsc{Partition Coloring Problem} with three parameters: the number of colors, the number of parts of the partition, and the maximum size of each part of the partition. Furthermore, we give a new exact algorithm for this problem.