Carving-decomposition based algorithms for the maximum path coloring problem

Given a set P of paths in a graph G and k colors, the maximum path coloring (Max-PC) problem is to find a maximum subset of P and assign a color to each path of the subset such that the paths with the same color are edge-disjoint. The Max-PC problem is an abstract model for many important routing problems including the all-optical routing. We give a carving-decomposition based exact algorithm for the Max-PC problem. A carving-decomposition of G is a system of edge-cut sets which decomposes G into subgraphs with each vertex of G a minimal subgraph. Our algorithm first finds a carving-decomposition of G and then solves the problem using the dynamic programming based on the carving-decomposition. We also give a 1.58-approximation algorithm for the Max-PC problem. Let L be the maximum number of paths in P on any edge of G and let γ be the maximum cardinality of any edge-cut in a given carving-decomposition. Our exact algorithm solves the Max-PC problem in O((L + 1)^1.5kγ n²) time and the approximation algorithm runs in O((L + 1)^1.5γ kn²) time for G of n vertices. Our algorithms can be used to solve the Max-PC problem on directed graphs as well. Our computational study shows that the exact algorithm can solve the Max-PC problem for small k and γ in a practical time and the approximation algorithm gives solutions close to the optimal ones for practical values of k and L on graphs with small γ such as rings.