Two-Class (r,k)-Coloring: Coloring with Service Guarantees

TL;DR

This paper studies a new graph coloring problem involving two classes of colors with different conflict rules, establishing its computational complexity and approximability limits.

Contribution

It introduces the Two-Class (r,k)-Coloring problem, proving NP-completeness and APX-completeness results, and explores its approximation boundaries.

Findings

NP-complete for all (r,k) except (0,2)

Cannot be approximated within any constant factor for certain parameters

APX-complete for k ≥ r ≥ 2

Abstract

This paper introduces the Two-Class ($r$,$k$)-Coloring problem: Given a fixed number of $k$ colors, such that only $r$ of these $k$ colors allow conflicts, what is the minimal number of conflicts incurred by an optimal coloring of the graph? We establish that the family of Two-Class ($r$,$k$)-Coloring problems is NP-complete for any $k \geq 2$ when $(r, k) \neq (0,2)$. Furthermore, we show that Two-Class ($r$,$k$)-Coloring for $k \geq 2$ colors with one ($r = 1$) relaxed color cannot be approximated to any constant factor ($\notin$ APX). Finally, we show that Two-Class ($r$,$k$)-Coloring with $k \geq r \geq 2$ colors is APX-complete.