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Minimum Manhattan Network is NPComplete
"... A rectilinear path between two points p,q ∈ R 2 is a path connecting p and q with all its line segments horizontal or vertical segments. Furthermore, a Manhattan path between p and q is a rectilinear path with its length exactly dist(p, q): = p.x − q.x  + p.y − q.y. Given a set T of n points in ..."
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A rectilinear path between two points p,q ∈ R 2 is a path connecting p and q with all its line segments horizontal or vertical segments. Furthermore, a Manhattan path between p and q is a rectilinear path with its length exactly dist(p, q): = p.x − q.x  + p.y − q.y. Given a set T of n points in R 2, a network G is said to be a Manhattan network on T, if for all p, q ∈ T there exists a Manhattan path between p and q with all its line segments in G. For the given point set T, the Minimum Manhattan Network (MMN) Problem is to find a Manhattan network G on T with the minimum network length. In this paper, we shall prove that the decision version of MMN is strongly NPcomplete, using the reduction from the wellknown 3SAT problem, which requires a number of gadgets. The gadgets have similar structures, but play different roles in simulating the 3SAT formula. The reduction has been implemented with a computer program.
Greedy construction of 2approximation minimum Manhattan network
 IN: PROCEEDINGS OF THE 19TH INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION
, 2008
"... Given a set T of n points in IR 2, a Manhattan Network G is a network with all its edges horizontal or vertical segments, such that for all p, q ∈ T, in G there exists a path (named a Manhattan path) of the length exactly the Manhattan distance between p and q. The Minimum Manhattan Network proble ..."
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Given a set T of n points in IR 2, a Manhattan Network G is a network with all its edges horizontal or vertical segments, such that for all p, q ∈ T, in G there exists a path (named a Manhattan path) of the length exactly the Manhattan distance between p and q. The Minimum Manhattan Network problem is to find a Manhattan network of the minimum length, i.e., the total length of the segments of the network is to be minimized. In this paper we present a 2approximation algorithm with time complexity O(n log n), which improves the 2approximation algorithm with time complexity O(n²). Moreover, compared with other 2approximation algorithms employing linear programming or dynamic programming technique, it was first discovered that only greedy strategy suffices to get 2approximation network.
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 NPcomplete discrete combinatorial optimization problem we employ a simple metaheuristic namely Simulated Annealing. We have also developed benchmark datas ..."
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In this paper we address the Minimum Manhattan Network (MMN) problem. It is an important geometric problem with vast applications. As it is an NPcomplete 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.