Strong Conflict-Free Coloring of Intervals

TL;DR

This paper introduces polynomial algorithms for k-strong conflict-free coloring of points with respect to intervals on a line, providing approximation guarantees for general and special cases, advancing solutions for conflict-free coloring problems.

Contribution

The paper presents the first polynomial algorithms with approximation guarantees for k-strong conflict-free coloring of points with respect to intervals.

Findings

Polynomial algorithms with approximation factors 5-2/k for general cases.

A 2-approximation algorithm for the case of all possible intervals.

Effective solutions for conflict-free coloring with multiple colors.

Abstract

We consider the k-strong conflict-free coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflict-free, in the sense that in every interval I there are at least k colors each appearing exactly once in I. In this paper, we present a polynomial algorithm for the general problem; the algorithm has an approximation factor 5-2/k when k\geq2 and approximation factor 2 for k=1. In the special case the family contains all the possible intervals on the given set of points, we show that a 2 approximation algorithm exists, for any k\geq1.