Balanced paths in acyclic networks: tractable cases and related approaches

Given a weighted acyclic network G and two nodes s and t in G, we consider the problem of computing k balanced paths from s to t, that is, k paths such that the difference in cost between the longest and the shortest path is minimized.The problem has several variants. We show that, whereas the general problem is solvable in pseudopolynomial time, both the arc-disjoint and the node-disjoint variants (i.e., the variants where the k paths are required to be arc-disjoint and node-disjoint, respectively) are strongly NP-Hard. We then address some significant special cases of such variants, and propose exact as well as approximate algorithms for their solution. The proposed approaches are also able to solve versions of the problem in which k origin-destination pairs are provided, and a set of k paths linking the origin-destination pairs has to be computed in such a way to minimize the difference in cost between the longest and the shortest path in the set.