Rainbow covering

Rainbow covering is a problem in computational geometry.

Definitions
There is a set J of n colored intervals on the real line, and a set P of points on the real line.

A subset Q of J is called a rainbow set if it contains at most a single interval of each color.

A set of intervals J is called a covering of P if each point in P is contained in at least one interval of Q.

The Rainbow covering problem is the problem of finding a rainbow set Q that is a covering of P.

Hardness
The problem is NP-hard. The proof is by reduction from linear SAT.

Generalization
Rainbow covering is a special case of the geometric conflict-free set cover problem.

In this problem, there is a set O of m closed geometric objects, and a conflict-graph GO on O.

A subset Q of O is called conflict-free if it is an independent set in GO, that is, no two objects in Q are connected by an edge in GO. A rainbow set is a conflict-free set in the special case in which GO is made of disjoint cliques, where each clique represents a color.

Geometric conflict-free set cover is the problem of finding a conflict-free subset of O that is a covering of P.

Banik, Panolan, Raman, Sahlot and Saurabh prove the following for the special case in which the conflict-graph has bounded arboricity:


 * If the geometric cover problem is fixed-parameter tractable (FPT), then the conflict-free geometric cover problem is FPT.
 * If the geometric cover problem admits an r-approximation algorithm, then the conflict-free geometric cover problem admits a similar approximation algorithm in FPT time.