Two-Class (r,k)-Coloring: Coloring with Service Guarantees
P\'al Andr\'as Papp, Roland Schmid, Valentin Stoppiello, Roger, Wattenhofer

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 (,)-Coloring problem: Given a fixed number of colors, such that only of these 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 (,)-Coloring problems is NP-complete for any when . Furthermore, we show that Two-Class (,)-Coloring for colors with one () relaxed color cannot be approximated to any constant factor ( APX). Finally, we show that Two-Class (,)-Coloring with colors is APX-complete.
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Taxonomy
Topicsgraph theory and CDMA systems · Scheduling and Timetabling Solutions
