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All-shortest-path 2-interval routing is NP-complete

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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. A fundamental problem is to characterize the networks that admit k-IRS. All 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 problems have been proved to be NP-complete for every k≥3, and remain open for k=1,2. In this paper, we close the open case of k=2 by showing that it is NP-complete to decide whether a graph admits an all-shortest-path 2-IRS. The same proof is also valid for all-shortest-path Strict 2-IRS. All-shortest-path Strict k-IRS is previously known to be polynomial for k=1, open for k=2,3, and NP-complete for every constant k≥4.

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Correspondence to Kai Wang.

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The research work of K. Wang was funded by the U.S. National Science Foundation under grant ASC070018T, by the U.S. South Dakota Governor’s Individual Research Seed Grant Award, by the South Dakota Brin Program of NIH/NCRR, and by the China National Science Foundation under grants 60673151 and 60403036.

The research work of R. Wang and Y. Liu was funded by the China National Science Foundation under grants 60673151 and 60403036.

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Wang, K., Wang, R. & Liu, Y. All-shortest-path 2-interval routing is NP-complete. J. Appl. Math. Comput. 32, 479–489 (2010). https://doi.org/10.1007/s12190-009-0265-2

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