Congestion-Free Rerouting of Flows on DAGs

We initiate the theoretical study a fundamental algorithmic problem: How to schedule the rerouting of k unsplittable network flows (of a certain demand) from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for k = 2 flows, which motivates us to study rerouting on a most fundamental family of flow graphs, namely DAGs. Interestingly, we find that for general k, deciding whether an unsplittable multi-commodity flow rerouting schedule exists, is NP-hard even on DAGs. Our NP-hardness proofs are non-trivial. We then present two polynomial-time algorithms to solve the route update problem for a constant number of flows on DAGs. Both algorithms employ a decomposition of the flow graph into smaller parts which we call blocks. Based on the given block decomposition, we define a dependency graph and use it to devise a deterministic algorithm for computing an optimal solution for k = 2 flows. For arbitrary but fixed k, we introduce a weaker notion of a dependency graph and present our main contribution: an elegant linear-time algorithm which solves the problem in time 2O(k log k)O(|G|). 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory