NP-completeness results for all-shortest-path interval routing

Abstract

k-Interval Routing Scheme (k-IRS) is a compact routing method that allows up to k interval labels to be assigned to an arc. A fundamental problem is to characterize the networks that admit k-IRS. Many of the problems related to single-shortest-path k-IRS have already been shown to be NP-complete. For all-shortest-path k-IRS, the characterization problem remains open for k ≥ 1. We investigate the time complexity of devising minimal-space all-shortest-path k-IRS and prove that it is NP-complete to decide whether a graph admits an all-shortest-path k-IRS, for every integer k ≥ 3, as well as whether a graph admits an all-shortest-path k-strict IRS, for every integer k ≥ 4. These are the first NP-completeness results for all-shortest-path k-IRS where k is a constant and the graph is unweighted. Moreover, the NP-completeness holds also for the linear case. © Springer-Verlag Berlin Heidelberg 2004.