The maximum disjoint paths problem on multi-relations social networks

Motivated by applications to social network analysis (SNA), we study the problem of finding the maximum number of disjoint unicolor paths in an edge-colored graph. We show the NP-hardness and the approximability of the problem, and both approximation and exact algorithms are proposed. Since short paths are much more significant in SNA, we also study the length-bounded version of the problem, in which the lengths of paths are required to be upper bounded by a fixed integer l. It is shown that the problem can be solved in polynomial time for l = 3 and is NP-hard for l ≥ 4. We also show that the problem can be approximated with ratio (l − 1)/2 + ε in polynomial time for any ε > 0. Particularly, for l = 4, we develop an efficient 2-approximation algorithm.