Minimum Manhattan Network Problem Research Articles

Dynamic Programming Approach to the Generalized Minimum Manhattan Network Problem

We study the generalized minimum Manhattan network (GMMN) problem: given a set $$P$$ of pairs of points in the Euclidean plane $${\mathbb{R}}^2$$ , we are required to find a minimum-length geometric network which consists of axis-aligned segments and contains a shortest path in the $$L_1$$ metric (a so-called Manhattan path) for each pair in $$P$$ . This problem commonly generalizes several NP-hard network design problems that admit constant-factor approximation algorithms, such as the rectilinear Steiner arborescence (RSA) problem, and it is open whether so does the GMMN problem. As a bottom-up exploration, Schnizler (2015) focused on the intersection graphs of the rectangles defined by the pairs in $$P$$ , and gave a polynomial-time dynamic programming algorithm for the GMMN problem whose input is restricted so that both the treewidth and the maximum degree of its intersection graph are bounded by constants. In this paper, as the first attempt to remove the degree bound, we provide a polynomial-time algorithm for the star case, and extend it to the general tree case based on an improved dynamic programming approach.

Approximating the Generalized Minimum Manhattan Network Problem

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 e )-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.

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, no two terminals on the same horizontal or vertical line, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) belong to the axis-parallel grid defined by T and are oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this article, we present a polynomial factor 2-approximation algorithm for the bidirected minimum Manhattan network problem. © 2016 Wiley Periodicals, Inc. NETWORKS, 2016

Optimal realizations of two-dimensional, totally-decomposable metrics

A realization 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 realization (G,w) is called optimal if the sum of its edge weights is minimal over all such realizations. Optimal realizations 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. A. Dress (1984) showed that the optimal realizations 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 realization 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 realizations of two-dimensional totally-decomposable metrics.

A Simulated Annealing approach for solving Minimum Manhattan Network Problem

In this paper we address the Minimum Manhattan Network (MMN) problem. It is an important geometric problem with vast applications. As it is an NP-complete discrete combinatorial optimization problem we employ a simple metaheuristic namely Simulated Annealing. We have also developed benchmark datasets and tested our algorithm with the dataset.

Approximating Minimum Manhattan Networks in Higher Dimensions

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in ${\mathbb {R}}^{d}$ , find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, L 1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\mathcal{P}} = \mathcal{NP}$ ). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ?>0, an O(n ? )-approximation algorithm. For 3D, we also give a 4(k?1)-approximation algorithm for the case that the terminals are contained in the union of k?2 parallel planes.

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

Let ${\mathcal{B}}$ be a centrally symmetric convex polygon of ℝ2 and ‖p−q‖ be the distance between two points p,q∈ℝ2 in the normed plane whose unit ball is ${\mathcal{B}}$. For a set T of n points (terminals) in ℝ2, a ${\mathcal{B}}$ -network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of ${\mathcal{B}}$ and for every pair of terminals t i and t j , the network N(T) contains a shortest ${\mathcal{B}}$ -path between them, i.e., a path of length ‖t i −t j ‖. A minimum ${\mathcal{B}}$ -network on T is a ${\mathcal{B}}$ -network of minimum possible length. The problem of finding minimum ${\mathcal{B}}$ -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball ${\mathcal{B}}$ 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. 393–402, 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 ${\mathcal{B}}$ -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.

GREEDY CONSTRUCTION OF 2-APPROXIMATE MINIMUM MANHATTAN NETWORKS

Given a set T of n points in ℝ2, a Manhattan network on T is a graph G = (V,E) with the property that all the edges in E are vertical or horizontal line segments connecting points in V ⊇ T and for all p, q ∈ T, the graph contains a path having the length exactly L1 distance between p and q. The Minimum Manhattan Network problem is to find a Manhattan network of minimum length, i.e. minimizing the total length of the line segments of the network. In this paper we present a 2-approximation algorithm with time complexity O(n log n), which improves over a recent combinatorial 2-approximation algorithm with running time O(n2). Moreover, compared with other 2-approximation algorithms using linear programming or dynamic programming techniques, we show that a greedy strategy suffices to obtain a 2-approximation algorithm.

Minimum Manhattan Network is NP-Complete

Given a set T of n points in ℝ2, a Manhattan network on T is a graph G with the property that for each pair of points in T, G contains a rectilinear path between them of length equal to their distance in the L 1-metric. The minimum Manhattan network problem is to find a Manhattan network of minimum length, i.e., minimizing the total length of the line segments in the network. In this paper, we prove that the decision version of the MMN problem is strongly NP-complete, using a reduction from the well-known 3-SAT problem, which requires a number of gadgets. The gadgets have similar structures, but play different roles in simulating a 3-CNF formula.

The Transitive Minimum Manhattan Subnetwork Problem in 3 dimensions

We consider the Minimum Manhattan Subnetwork (MMSN) Problem which generalizes the already known Minimum Manhattan Network (MMN) Problem: Given a set P of n points in the plane, find shortest rectilinear paths between all pairs of points. These paths form a network, the total length of which has to be minimized. From a graph theoretical point of view, a MMN is a 1-spanner with respect to the L 1 metric. In contrast to the MMN problem, a solution to the MMSN problem does not demand L 1 -shortest paths for all point pairs, but only for a given set R ⊆ P × P of pairs. The complexity status of the MMN problem is still unsolved in ≥2 dimensions, whereas the MMSN was shown to be N P -complete considering general relations R in the plane. We restrict the MMSN problem to transitive relations R T ( Transitive Minimum Manhattan Subnetwork (TMMSN) Problem) and show that the TMMSN problem in 3 dimensions is N P -complete.

A rounding algorithm for approximating minimum Manhattan networks

For a set T of n points (terminals) in the plane, a Manhattan network on T is a network N ( T ) = ( V , E ) with the property that its edges are horizontal or vertical segments connecting points in V ⊇ T and for every pair of terminals, the network N ( T ) contains a shortest l 1 -path between them. A minimum Manhattan network on T is a Manhattan network of minimum possible length. The problem of finding minimum Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan [J. Gudmundsson, C. Levcopoulos, G. Narasimhan, Approximating a minimum Manhattan network, Nordic Journal of Computing 8 (2001) 219–232. Proc. APPROX’99, 1999, pp. 28–37] and its complexity status is unknown. Several approximation algorithms (with factors 8, 4, and 3) have been proposed; recently Kato, Imai, and Asano [R. Kato, K. Imai, T. Asano, An improved algorithm for the minimum Manhattan network problem, ISAAC’02, in: LNCS, vol. 2518, 2002, pp. 344–356] have given a factor 2-approximation algorithm, however their correctness proof is incomplete. In this paper, we propose a rounding 2-approximation algorithm based on an LP-formulation of the minimum Manhattan network problem.

The minimum Manhattan network problem

Given a set of points in the plane and a constant t>=1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two ...

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t ⩾ 1 , a Euclidean t- spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively. In this paper we study 1-spanners under the Manhattan (or L 1 -) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O ( n log n ) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P. We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.