Path coloring on the mesh

In the minimum path coloring problem, we are given list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al and Raghavan and Upfal as a model for routing in all-optical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N) approximation result of Kleinberg and Tardos for path coloring on the N×N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sub-logarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integrality. They are motivated by the method used by Leighton, Maggs, and Rao for packet routing.