Minimum Manhattan network is NP-complete

Chin, F.; Guo, Zeyu; Sun, He

doi:10.1145/1542362.1542429

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…They also conjectured that there exists a 2-approximation algorithm for this problem and asked if this problem is NP-complete. Quite recently, Chin, Guo, and Sun [6] solved this last open question from [16] and established that indeed the minimum Manhattan network problem is strongly NP-complete. Kato, Imai, and Asano [18] presented a 2-approximation algorithm, however, their correctness proof is incomplete (see [1]).…”

Section: Known Resultsmentioning

confidence: 99%

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Catusse¹,

Chepoi²,

Nouioua³

et al. 2011

Algorithmica

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Let B be a centrally symmetric convex polygon of R 2 and p − q be the distance between two points p, q ∈ R 2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R 2 , a B-network on T is a network N(T ) = (V , E) with the property that its edges are parallel to the directions of B and for every pair of terminals t i and t j , the network N(T ) contains a shortest B-path between them, i.e., a path of length t i − t j . A minimum B-network on T is a B-network of minimum possible length. The problem of finding minimum B-networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance p − q is the l 1 or the l ∞ -distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. [393][394][395][396][397][398][399][400][401][402] 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum B-network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56-69, 2008, and APPROX-RANDOM, pp. 40-51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem. N. Catusse · V. Chepoi ( ) · K. Nouioua · Y. Vaxès

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Section: Known Resultsmentioning

confidence: 99%

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Catusse¹,

Chepoi²,

Nouioua³

et al. 2011

Algorithmica

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“…A solution of RSSG is a rectilinear routing containing all master-to-slave rectilinear shortest paths, with total wire length as small as possible. RSSG is a generalization of the minimum rectilinear Steiner arborescence (RSA) problem [8][13] and a relaxation of the Minimum Manhattan network (MMN) problem [3], both of which have been proven to be NP-Complete in [14] and [7] respectively. The RSA problem corresponds to the case where there is only one master s. We will introduce two approximation algorithms for RSSG in the following subsections.…”

Section: Problem Two: Approximation Al-gorithms For Generating Recti-mentioning

confidence: 99%

Low-power gated bus synthesis for 3d ic via rectilinear shortest-path steiner graph

Cheng

Kahng

et al. 2012

Proceedings of the 2012 ACM International Symposium on International Symposium on Physical Design

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In this paper, we propose a new approach for gated bus synthesis [16] with minimum wire capacitance per transaction in three-dimensional (3D) ICs. The 3D IC technology connects different device layers with through-silicon vias (TSV), which need to be considered differently from metal wire due to reliability issues and a larger footprint. Practically, the number of TSVs is bounded between layers; thus, we first devise dynamic programming and local search techniques to determine the optimal TSV locations. We then employ two approximation algorithms to generate a rectilinear shortestpath Steiner graph in each device layer. One algorithm extends the well-known greedy heuristic for the Rectilinear Steiner Arborescence problem and handles large cases with high efficiency. The other algorithm utilizes a linear programming relaxation and rounding technique which costs more time and generates a nearly-optimal Steiner graph. Experimental results show that our algorithms can construct shortest-path Steiner graphs with 22% less total wire length than the previous method of Wang et al. [16].

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Optimal realizations of two-dimensional, totally-decomposable metrics

Herrmann

Koolen

Lesser

et al. 2015

Discrete Mathematics

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A realisation of a metric d on a finite set X is a weighted graph (G, w) whose vertex set contains X such that the shortest-path distance between elements of X considered as vertices in G is equal to d. Such a realisation (G, w) is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations always exist, although it is NP-hard to compute them in general, and they have applications in areas such as phylogenetics, electrical networks and internet tomography. In [Adv. in Math. 53, 1984, 321-402] A. Dress showed that the optimal realisations of a metric d are closely related to a certain polytopal complex that can be canonically associated to d called its tight-span. Moreover, he conjectured that the (weighted) graph consisting of the zero-and one-dimensional faces of the tight-span of d must always contain an optimal realisation as a homeomorphic subgraph. In this paper, we prove that this conjecture does indeed hold for a certain class of metrics, namely the class of totally-decomposable metrics whose tight-span has dimension two. As a corollary, it follows that the minimum Manhattan network problem is a special case of finding optimal realisations of two-dimensional totally-decomposable metrics.

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Minimum Manhattan network is NP-complete

Cited by 5 publications

References 13 publications

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Low-power gated bus synthesis for 3d ic via rectilinear shortest-path steiner graph

Optimal realizations of two-dimensional, totally-decomposable metrics

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