Minimum Manhattan Network is NP-Complete

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

doi:10.1007/s00454-011-9342-z

Cited by 18 publications

(11 citation statements)

References 11 publications

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“…and O(log n + √ r) query time, where is an arbitrarily small positive constant and r is an arbitrary integer, such that 1 < r < n. Recently, Amani et al [15] show how to compute a plane problem of finding a 1-spanner in L 1 -metric [9]. Given a rectilinear polygon with n vertices, in linear time, Schuierer [16] constructs a data structure that can report the shortest path (in L 1 -metric) for any pair of query points in that polygon in O(log +k) time where k is the number of segments in the shortest path.…”

Section: (C)mentioning

confidence: 99%

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Linear Size Planar Manhattan Network for Convex Point Sets

Jana¹,

Maheshwari²,

Roy³

2019

Preprint

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Let G = (V, E) be an edge weighted geometric graph such that every edge is horizontal or vertical. The weight of an edge uv ∈ E is its length. Let W G (u, v) denote the length of a shortest path between a pair of vertices u and v in G. The graph G is said to be a Manhattan network for a given point set P in the plane if P ⊆ V and ∀p, q ∈ P , W G (p, q) = pq 1 . In addition to P , graph G may also include a set T of Steiner points in its vertex set V . In the Manhattan network problem, the objective is to construct a Manhattan network of small size for a set of n points. This problem was first considered by Gudmundsson et al. [1]. They give a construction of a Manhattan network of size Θ(n log n) for general point set in the plane. We say a Manhattan network is planar if it can be embedded in the plane without any edge crossings. In this paper, we construct a linear size planar Manhattan network for convex point set in linear time using O(n) Steiner points. We also show that, even for convex point set, the construction in Gudmundsson et al. [1] needs Ω(n log n) Steiner points and the network may not be planar.

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Section: (C)mentioning

confidence: 99%

“…Also, there are O(n log n) time approximation algorithms with factors 8 [1], 3 [7], and 2 [8]. Recently Chin et al [9] proved that the decision version of the MMN problem is strongly NP-complete. Recently, Knauer et al [10] showed that this problem is fixed parameter tractable.…”

Section: Introductionmentioning

confidence: 99%

Linear Size Planar Manhattan Network for Convex Point Sets

Jana¹,

Maheshwari²,

Roy³

2019

Preprint

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“…First note that, in view of the fact that the problem of computing a minimum Manhattan network is NP-hard [7] and the fact that there is always a minimum Manhattan network that is contained in the grid induced by the given point set (see e.g. [3]), we have: Proposition 6.…”

Section: Finding Minimal Subrealizationsmentioning

confidence: 99%

“…First, we consider a special instance of the problem where we take metrics to be l 1 -distances between points in the plane. In Section 5 we show that finding optimal realizations of such a metric D in G D is equivalent to the so-called minimum Manhattan network problem (which was also recently shown to be NP-hard [7]). This allows us to compare the realizations computed by our heuristic with realizations computed using a mixed integer linear program (MIP) for the minimum Manhattan network problem presented in [3] (see also [26] for a comprehensive list of references on other approaches for solving this well-studied problem).…”

Section: Introductionmentioning

confidence: 99%

Searching for realizations of finite metric spaces in tight spans

Herrmann

Moulton

Spillner

2013

Discrete Optimization

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An important problem that commonly arises in areas such as internet trafficflow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the metric D and its so-called tight span T D . The tight span T D is a canonical polytopal complex that can be associated to D, and our approach explores parts of T D for realizations in a way that is similar to the classical simplex algorithm. We also provide computational results illustrating the performance of our approach for different types of metrics, including l 1 -distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.

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“…For example, Wendell and Hurter [27] establish this type of results for the Weber problem (the weighted version of the median problem) while Hansen, Perreur, and Thisse [15] proved a similar result for the NP-hard multifacility location problem. In [6], it was shown that Pareto envelopes in the l 1 -plane contain at least one minimum Manhattan network, and this fact was used in all factor 2 approximation algorithms [6,12,14,19] for the NP-hard [7] minimum Manhattan network problem. For other results in this vein, see [23].…”

Section: Introductionmentioning

confidence: 99%

Pareto Envelopes in Simple Polygons

Chepoi

Nouioua

Thiel

et al. 2010

Int. J. Comput. Geom. Appl.

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For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t′ ∈ T such that d(x, t′) < d(y, t′). The set of non-dominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (ℝ2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of point-sets in simple polygons P endowed with geodesic d2- or d1-metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhn's characterization extends to Pareto envelopes in simple polygons with d2-metric, while that of Chalmet et al. extends to simple rectilinear polygons with d1-metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes.

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Minimum Manhattan Network is NP-Complete

Cited by 18 publications

References 11 publications

Linear Size Planar Manhattan Network for Convex Point Sets

Linear Size Planar Manhattan Network for Convex Point Sets

Searching for realizations of finite metric spaces in tight spans

Pareto Envelopes in Simple Polygons

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