On Fixed-Parameter Tractability of Some Routing Problems

Disjoint Paths is the problem of finding paths between given pairs of terminals in a graph such that no vertices are shared between paths. We analyze fixed-parameter tractability of several new Disjoint Paths-like routing problems motivated by congestion control in computer networks. In one model we are interested in finding paths between k pairs of terminals such that the first edge of each path is not shared with any other path. We prove that this problem is fixed-parameter tractable on directed graphs, in contrast to Disjoint Paths that are known to be NP-hard even for k = 2. We improve our algorithm for two special cases: when the graph is acyclic and when all sources lie in distinct nodes. We consider extensions: a second-node-disjoint analog and a slightly generalized version of SAT. Another model, bottleneck-edge-disjoint paths, is a generalization of Disjoint Paths. For directed acyclic graphs, we show that bottleneck-edge-disjoint paths is W [1]-hard and hence unlikely to be fixedparameter tractable. We give an algorithm that runs in time nO(k). These two results easily extend to Unsplittable Flows.