Line system design and a generalized coloring problem

Abstract

We study a generalized coloring and routing problem for interval and circular graphs that is motivated by design of optical line systems. In this problem we are interested in rinding a coloring and routing of "demands" of minimum total cost where the total cost is obtained by accumulating the cost incurred at certain "links" in the graph. The colors are partitioned in sets and the sets themselves are ordered so that colors in higher sets cost more. The cost of a "link" in a coloring is equal to the cost of the most expensive set such that a demand going through the link is colored with a color in this set. We study different versions of the problem and characterize their complexity by presenting tight upper and lower bounds. For the interval graph we show that the most general problem is hard to approximate to within √s and we complement this result with a O(√s)-approximation algorithm for the problem. Here s is proportional to the number of color sets. For the circular graph problem we show that most versions of the problem are hard to approximate to any bounded ratio and we present a 2(1 + ε) approximation scheme for a special version of the problem. © Springer-Verlag 2003.