On the Tractability of (k,i)-Coloring

TL;DR

This paper investigates the computational complexity and algorithms for the (k,i)-coloring problem in graphs, providing a parameterized algorithm, an improved exact algorithm for specific cases, and proving NP-completeness for fixed parameters.

Contribution

It introduces a parameterized algorithm based on feedback vertex set size, offers a faster exact algorithm for (k, k-1)-coloring, and proves NP-completeness for fixed i, k, resolving several open questions.

Findings

Parameterizes (k,i)-coloring by feedback vertex set size with a new algorithm.

Provides a faster, simpler exact algorithm for (k, k-1)-coloring.

Proves NP-completeness of (k,i)-coloring for fixed i, k, settling longstanding conjectures.

Abstract

In an undirected graph, a proper (k,i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k,i)-coloring problem is to compute the minimum number of colors required for a proper (k,i)-coloring. This is a generalization of the classic graph coloring problem. We show a parameterized algorithm for the (k,i)-coloring problem with the size of the feedback vertex set as a parameter. Our algorithm does not use tree-width machinery, thus answering a question of Majumdar, Neogi, Raman and Tale [CALDAM 2017]. We also give a faster and simpler exact algorithm for (k, k-1)-coloring. From the hardness perspective, we show that the (k,i)-coloring problem is NP-complete for any fixed values i, k, whenever i<k, thereby settling a conjecture of Mendez-Diaz and Zabala [1999] and again asked by Majumdar, Neogi, Raman and Tale. The NP-completeness result improves the partial NP-completeness shown in the preliminary version of this paper published in CALDAM 2018.