Two-Class (r,k)-Coloring: Coloring with Service Guarantees
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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 ≥ 2 when (r, k) ≠ (0,2). Furthermore, we show that Two-Class (r,k)-Coloring for k ≥ 2 colors with one (r = 1) relaxed color cannot be approximated to any constant factor (∉ APX). Finally, we show that Two-Class (r,k)-Coloring with k ≥ r ≥ 2 colors is APX-complete.
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