Abstract
We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in ℝ2. The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an O(logn)-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple O(logd + 1 n)-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(n ε)-ratio for MMN in d dimensions [ESA’11]. En route, we show that an existing O(logn)-approximation algorithm for 2D-RSA generalizes to higher dimensions.
This work was supported by the ESF EuroGIGA project GraDR (DFG grant Wo 758/5-1).
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Das, A., Fleszar, K., Kobourov, S., Spoerhase, J., Veeramoni, S., Wolff, A. (2013). Approximating the Generalized Minimum Manhattan Network Problem. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_67
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