Approximating the generalized minimum Manhattan network problem

Aparna Das, Krzysztof Fleszar, Stephen Kobourov, Joachim Spoerhase, Sankar Veeramoni, Alexander Wolff

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

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(log d+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.

Original languageEnglish (US)
Title of host publicationAlgorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings
Pages722-732
Number of pages11
DOIs
StatePublished - Dec 1 2013
Event24th International Symposium on Algorithms and Computation, ISAAC 2013 - Hong Kong, China
Duration: Dec 16 2013Dec 18 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8283 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other24th International Symposium on Algorithms and Computation, ISAAC 2013
CountryChina
CityHong Kong
Period12/16/1312/18/13

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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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 Algorithms and Computation - 24th International Symposium, ISAAC 2013, Proceedings (pp. 722-732). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8283 LNCS). https://doi.org/10.1007/978-3-642-45030-3_67