Complexity and online algorithms for minimum skyline coloring of intervals

Abstract

Graph coloring has been studied extensively in the literature. The classical problem concerns the number of colors used. In this paper, we focus on coloring intervals where the input is a set of intervals and two overlapping intervals cannot be assigned the same color. In particular, we are interested in the setting where there is an increasing cost associated with using a higher color index. Given a set of intervals (on a line) and a coloring, the cost of the coloring at any point is the cost of the maximum color index used at that point and the cost of the overall coloring is the integral of the cost over all points on the line. The objective is to assign a valid color to each interval and minimize the total cost of the coloring. Intuitively, the maximum color index used at each point forms a skyline and so the objective is to obtain a minimum skyline coloring. The problem arises in various applications including optical networks and job scheduling. Alicherry and Bhatia defined in 2003 a more general problem in which the colors are partitioned into classes and the cost of a color depends solely on its class. This problem is NP-hard and the reduction relies on the fact that some color class has more than one color. In this paper we show that when each color class only contains one color, this simpler setting remains NP-hard via a reduction from the arc coloring problem. In addition, we initiate the study of the online setting and present an asymptotically optimal online algorithm. We further study a variant of the problem in which the intervals are already partitioned into sets and the objective is to assign a color to each set such that the total cost is minimum. We show that this seemingly easier problem remains NP-hard by a reduction from the optimal linear arrangement problem.