Short length Menger's theorem and reliable optical routing

Abstract

In the minimum path coloring problem, we are given a graph and a set of pairs of vertices of the graph and we are asked to connect the pairs by colored paths in such a way that paths of the same color are edge disjoint. In this paper we deal with a generalization of this problem where we are asked to connect each pair by k edge disjoint paths of the same color. The objective is to minimize the number of colors. The reason for multiple paths between the same pair of vertices is to ensure fault tolerance of the connections. We propose an O ( k 2 F ) = O ( k 2 Δ α - 1 log n ) approximation algorithm for this problem where F is the flow number of the graph, Δ is the maximum degree and α is the expansion. This is an improvement even for the special case k = 1 where, to our knowledge, the best previously known bound is weaker by a factor of log n . The underlying problem is that of finding several disjoint paths between a given pair of vertices. Menger's theorem provides a necessary and sufficient condition for the existence of k such paths. However, it does not say anything about the length of the paths although in communication problems the number of links used is an issue. We show that any two k -connected vertices are connected by k edge disjoint paths of average length O ( kF ) which improves an earlier result of Galil and Yu (Proceedings of the 27th Annual ACM Symposium on Theory of Computing, 1995) for several classes of graphs. In fact, this is only a corollary of a stronger result for multicommodity flow on networks with unit edge capacities: any multicommodity flow with k units for each commodity can be rerouted such that the flow for each commodity is shipped through k -tuples of edge disjoint paths of average length O ( kF ) without exceeding the edge capacities significantly.