On the hardness of minimizing space for all-shortest-path interval routing schemes

Authors:
Rui Wang;Francis C. M. Lau;Yan Yan Liu
Affiliations:
Department of Computer Science, The University of Hong Kong, Hong Kong and Department of Computer Science, The Ocean University of China, China;Department of Computer Science, The University of Hong Kong, Hong Kong;Department of Computer Science, The Ocean University of China, China
Venue:
Theoretical Computer Science
Year:
2007

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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; and global k-IRS allows not more than a total of k interval labels in the whole network. A fundamental problem is to characterize the networks that admit k-IRS (or global 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. In this paper, we study the time complexity of devising minimal-space all-shortest-path k-IRSs and show that it is NP-complete to decide whether a graph admits an all-shortest-path k-IRS, for every integer k=3, and so is that of deciding 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. The NP-completeness holds also for the linear case. We also prove that it is NP-complete to decide whether an unweighted graph admits an all-shortest-path IRS with global compactness of at most k, which also holds for the linear and strict cases.