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Approximation Algorithms for the MinimumLength Corridor and Related Problems
, 2007
"... Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the bou ..."
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Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the boundary of the rectangles, and must include at least one point from the boundary of every rectangle and from the rectangular boundary. The MLCR problem has been shown to be NPhard. In this paper we present the first polynomial time constant ratio approximation algorithm for the MLCR and MLCn problems. The MLCn problem is a generalization of the the MLCR problem where the rectangles are rectilinear kgons, for k ≤ n. We also present a polynomial time constant ratio approximation algorithm for the Group Traveling Salesperson Problem (GTSP) for a rectangle partitioned into rectilinear kgons as in the MLCn problem.
On the complexity of guarding problems on orthogonal arrangements
 Abstracts of the 20th Fall Workshop on Computational Geometry
, 2010
"... Consider a guard checking on some corridors in a building, the guard does not need to walk the entire length of a corridor, but visits at least one point of a corridor and looks up and down. What is the optimum solution for the guard’s tour? The corridors in a building can be modeled as a system of ..."
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Consider a guard checking on some corridors in a building, the guard does not need to walk the entire length of a corridor, but visits at least one point of a corridor and looks up and down. What is the optimum solution for the guard’s tour? The corridors in a building can be modeled as a system of connected orthogonal arrangement of vertical and horizontal line segments. The optimum problem is transformed into finding the shortest closed path along the line segments touching each line segment at least one point. Since it is a traveling salesmantype problem, we also consider the minimum spanning tree problem in the same model: finding the shortest tree along the line segments touching all line segments. We denote the first problem as a corridorTSP problem, and denote the second problem as a corridorMST problem, as shown in Figure 1 below. In this note we claim that these problems are both NPcomplete. These two problems belong to minimum length connection problems, which are presented in Canadian Conference of Computational Geometry 2000 [1]. Some corridor problems have been studied in [2] [3].
Connected Vertex Cover in 2Connected Planar Graph with Maximum Degree 4 is NPcomplete
 INTERNATIONAL JOURNAL OF MATHEMATICAL, PHYSICAL AND ENGINEERING SCIENCES
"... This paper proves that the problem of finding connected vertex cover in a 2connected planar graph ( CVC2) with maximum degree 4 is NPcomplete. The motivation for proving this result is to give a shorter and simpler proof of NPCompleteness of TRAMLC (the Top Right Access point MinimumLength Co ..."
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This paper proves that the problem of finding connected vertex cover in a 2connected planar graph ( CVC2) with maximum degree 4 is NPcomplete. The motivation for proving this result is to give a shorter and simpler proof of NPCompleteness of TRAMLC (the Top Right Access point MinimumLength Corridor) problem [1], by finding the reduction from CVC2. TRAMLC has many applications in laying optical fibre cables for data communication and electrical wiring in floor plans.The problem of finding connected vertex cover in any planar graph ( CVC) with maximum degree 4 is NPcomplete [2]. We first show that CVC2 belongs to NP and then we find a polynomial reduction from CVC to CVC2. Let a graph G0 and an integer K form an instance of CVC, where G0 is a planar graph and K is an upper bound on the size of the connected vertex cover in G0. We construct a 2connected planar graph, say G, by identifying the blocks and cut vertices of G0, and then finding the planar representation of all the blocks of G0, leading to a plane graph G1. We replace the cut vertices with cycles in such a way that the resultant graph G is a 2connected planar graph with maximum degree 4. We consider L = K − 2t +3 di where t is the number i=1 of cut vertices in G1 and di is the number of blocks for which i th cut vertex is common. We prove that G will have a connected vertex cover with size less than or equal to L if and only if G0 has a connected vertex cover of size less than or equal to K.
An Alternative Proof for the NPcompleteness of Top Right Access pointMinimum Length Corridor Problem
"... In the Top Right Access point Minimum Length Corridor (TRAMLC) problem [1], a rectangular boundary partitioned into rectilinear polygons is given and the problem is to find a corridor of least total length and it must include the top right corner of the outer rectangular boundary. A corridor is a t ..."
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In the Top Right Access point Minimum Length Corridor (TRAMLC) problem [1], a rectangular boundary partitioned into rectilinear polygons is given and the problem is to find a corridor of least total length and it must include the top right corner of the outer rectangular boundary. A corridor is a tree containing a set of line segments lying along the outer rectangular boundary and/or on the boundary of the rectilinear polygons. The corridor must contain at least one point from the boundaries of the outer rectangle and also the rectilinear polygons. Gutierrez and Gonzalez [1] proved that the MLC problem, along with some of its restricted versions and variants, are NPcomplete. In this paper, we give a shorter proof of NPCompleteness of TRAMLC by findig the reduction in the following way. Connected vertex cover in 2connected planar graph with maximum
Approximating Corridors and Tours via Restriction and Relaxation Techniques
, 2010
"... Abstract. Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/ ..."
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Abstract. Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the boundary of the rectangles, and must include at least one point from the boundary of every rectangle and from the rectangular boundary. The MLCR problem is known to be NPhard. We present the first polynomialtime constant ratio approximation algorithm for the MLCR and MLCk problems. The MLCk problem is a generalization of the MLCR problem where the rectangles are rectilinear cgons, for c ≤ k and k is a constant. We also present the first polynomialtime constant ratio approximation algorithm for the Group Traveling Salesperson Problem (GTSP) for a rectangular boundary partitioned into rectilinear cgons as in the MLCk problem. Our algorithms are based on the restriction and relaxation approximation techniques.
Hitting Set, Spanning Trees, and the Minimum Length Corridor Problem
, 2008
"... 1 Introduction We consider a problem that is a combination of hitting set and minimum spanning trees that generalizes the minimum length corridor problem. In the minimum length corridor problem, we are given a rectangle aligned with the axes in the plane that is subdivided into smaller rectangles al ..."
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1 Introduction We consider a problem that is a combination of hitting set and minimum spanning trees that generalizes the minimum length corridor problem. In the minimum length corridor problem, we are given a rectangle aligned with the axes in the plane that is subdivided into smaller rectangles also aligned with the axes in the plane. This defines a planar graph whose vertices are the corners of the rectangles and whose edges are segments of the sides joining them, where the edges have edge lengths. The aim is to find a subtree of this graph that touches all the faces, including the outer face, in at least one vertex, and of minimal total edge length.