The interval constrained 3-coloring problem

TL;DR

This paper proves that the interval constrained 3-coloring problem is NP-complete, establishing its computational difficulty even with a fixed number of colors, and shows the challenge in satisfying most constraints in feasible instances.

Contribution

It settles the open complexity question by proving NP-completeness for three colors and demonstrates the difficulty of nearly satisfying constraints in feasible cases.

Findings

NP-complete for three colors

Polynomial solvability for two or fewer colors

Difficulty in satisfying almost all constraints

Abstract

In this paper, we settle the open complexity status of interval constrained coloring with a fixed number of colors. We prove that the problem is already NP-complete if the number of different colors is 3. Previously, it has only been known that it is NP-complete, if the number of colors is part of the input and that the problem is solvable in polynomial time, if the number of colors is at most 2. We also show that it is hard to satisfy almost all of the constraints for a feasible instance.