Approximation Algorithms for Finding a Minimum Perimeter Polygon Intersecting a Set of Line Segments

2009 ◽  
pp. 363-374 ◽  
Author(s):  
Farzad Hassanzadeh ◽  
David Rappaport
Keyword(s):  
Approximation Algorithms ◽  
Line Segments ◽  
Minimum Perimeter

THE MINIMUM GUARDING TREE PROBLEM

2014 ◽  
Vol 06 (01) ◽  
pp. 1450011 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
JOSEPH S. B. MITCHELL ◽  
PAWEL ŻYLIŃSKI
Keyword(s):  
Approximation Algorithms ◽  
Nonempty Subset ◽  
Parallel Line ◽  
Minimum Length ◽  
Np Hard ◽  
Orthogonal Axis ◽  
Line Segments ◽  
Parallel Lines ◽  
Simple Alternative ◽  
Axis Parallel

Given a set ℒ of non-parallel lines in the plane and a nonempty subset ℒ′ ⊆ ℒ, a guarding tree for ℒ′ is a tree contained in the union of the lines in ℒ such that if a mobile guard (agent) runs on the edges of the tree, all lines in ℒ′ are visited by the guard. Similarly, given a connected arrangement 𝒮 of line segments in the plane and a nonempty subset 𝒮′ ⊆ 𝒮, we define a guarding tree for 𝒮′. The minimum guarding tree problem for a given set of lines or line segments is to find a minimum-length guarding tree for the input set. We provide a simple alternative (to [N. Xu, Complexity of minimum corridor guarding problems, Inf. Process. Lett.112(17–18) (2012) 691–696.]) proof of the problem of finding a guarding tree of minimum length for a set of orthogonal (axis-parallel) line segments in the plane. Then, we present two approximation algorithms with factors 2 and 3.98, respectively, for computing a minimum guarding tree for a subset of a set of n arbitrary non-parallel lines in the plane; their running times are O(n8) and O(n6 log n), respectively. Finally, we show that this problem is NP-hard for lines in 3-space.

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

LABELING A RECTILINEAR MAP WITH SLIDING LABELS

2001 ◽  
Vol 11 (02) ◽  
pp. 167-179 ◽  
Author(s):  
SUNG KWON KIM ◽  
CHAN-SU SHIN ◽  
TAE-CHEON YANG
Keyword(s):  
Approximation Algorithms ◽  
Finite Number ◽  
Vertical Line ◽  
Line Segments ◽  
Sliding Labels

A rectilinear map consists of a set of mutually non-intersecting rectilinear (i.e., horizontal or vertical) line segments, and each segment is allowed to use a rectangular label of height B and length the same as the segment. Sliding labels are not restricted to any finite number of predefined positions but can slide and be placed at any position as long as it intersects the segment. This paper considers three versions of the problem of labeling a rectilinear map with sliding labels and presents efficient exact and approximation algorithms for them.

GREEDY CONSTRUCTION OF 2-APPROXIMATE MINIMUM MANHATTAN NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 331-350 ◽  
Author(s):  
ZEYU GUO ◽  
HE SUN ◽  
HONG ZHU
Keyword(s):  
Dynamic Programming ◽  
Linear Programming ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Time Complexity ◽  
Minimum Length ◽  
Horizontal Line ◽  
Line Segments ◽  
Greedy Strategy ◽  
Programming Techniques

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.

scholarly journals THE TRAVELING SALESMAN PROBLEM FOR LINES AND RAYS IN THE PLANE

2012 ◽  
Vol 04 (04) ◽  
pp. 1250044 ◽  
Author(s):  
ADRIAN DUMITRESCU
Keyword(s):  
Approximation Algorithms ◽  
Traveling Salesman Problem ◽  
Shortest Path ◽  
Linear Time ◽  
Traveling Salesman ◽  
Tight Bound ◽  
Time Approximation ◽  
Minimum Perimeter ◽  
Open Curve ◽  
The Traveling Salesman Problem

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.

Sign in / Sign up

Export Citation Format

Share Document