A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics

Chan, T.-H. Hubert; Elbassioni, Khaled

doi:10.1007/s00454-011-9337-9

Cited by 23 publications

(3 citation statements)

References 25 publications

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“…The geometric TSP has a PTAS ( [3,23]), as does the TSPN in certain special cases (e.g., if the regions are "fat" and (weakly) disjoint [12,16,26]). Further, there is a quasipolynomial-time approximation scheme (QPTAS) for geometric instances of TSPN in any fixed dimension, and metric spaces of bounded doubling dimension, for the case of fat, (weakly) disjoint regions [5]. The TSPN for arbitrary connected regions in the plane has an O(log n)-approximation [18,22]; if the regions are (weakly) disjoint, there is a recent O(1)-approximation [27].…”

Section: Prior and Relatedmentioning

confidence: 99%

Approximating Watchman Routes

Mitchell

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given a connected polygonal domain P , the watchman route problem is to compute a shortest path or tour for a mobile guard (the "watchman") that is required to see every point of P . While the watchman route problem is polynomially solvable in simple polygons, it is known to be NP-hard in polygons with holes.We present the first polynomial-time approximation algorithm for the watchman route problem in polygonal domains.Our algorithm has an approximation factor O(log 2 n). Further, we prove that the problem cannot be approximated in polynomial time to within a factor of c log n, for a constant c > 0, assuming that P =NP. IntroductionA classic problem in computational geometry is the watchman route problem (WRP): Compute a shortest path/tour within a polygonal domain (polygon with holes) P so that every point of P is seen from some point of the path/tour, i.e., compute a shortest "visibility coverage tour". The WRP models a natural problem in robotics, in which a mobile robot/camera is to do a visual inspection of a domain or a part, whose geometry is given. Here, we focus on the offline setting, in which the geometry of the domain P is given; the problem has been studied extensively, both in the offline and online setting (see the surveys [24,25]).In this paper we give the first polynomial-time approximation algorithm for the WRP in polygonal domains in the plane. Our approximation factor is O(log 2 n); we also give the first hardness of approximation result for the WRP, showing that the problem cannot be approximated in polynomial time to within a factor of c log n, for a constant c > 0, assuming that P =NP. Here, n is the number of vertices of P .

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Section: Prior and Relatedmentioning

confidence: 99%

Approximating Watchman Routes

Mitchell

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…(1−tan α tan θ) 5 . Lemma 4 relies on the observation that the ratio of the volume of two cones is relative to the ratio of the area of their largest inscribed triangles (see appendix 10.12 for details).…”

Section: Lemma 10 Given Two Intersecting Conesmentioning

confidence: 99%

“…Many variants of TSPN in 3D are open. A PTAS was provided in [4] for disjoint polygons of comparable size and a QuasiPolynomial Time Approximation Scheme (QPTAS) in [5] for α-fat, weakly-disjoint neighborhoods. Recently in [14], we presented a polynomial time approximation algorithm for the 3D TSPN with intersecting neighborhoods when the neighborhoods take the form of right angular cones and their apex points lie on a planar surface.…”

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confidence: 99%

Approximation algorithms for tours of orientation-varying view cones

Stefas

Plonski

Isler

2020

The International Journal of Robotics Research

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This article considers the problem of finding a shortest tour to visit viewing sets of points on a plane. Each viewing set is represented as an inverted view cone with apex angle [Formula: see text] and height [Formula: see text]. The apex of each cone is restricted to lie on the ground plane. Its orientation angle (tilt) [Formula: see text] is the angle difference between the cone bisector and the ground plane normal. This is a novel variant of the 3D Traveling Salesman Problem with Neighborhoods (TSPN) called Cone-TSPN. One application of Cone-TSPN is to compute a trajectory to observe a given set of locations with a camera: for each location, we can generate a set of cones whose apex and orientation angles [Formula: see text] and [Formula: see text] correspond to the camera’s field of view and tilt. The height of each cone [Formula: see text] corresponds to the desired resolution. Recently, Plonski and Isler presented an approximation algorithm for Cone-TSPN for the case where all cones have a uniform orientation angle of [Formula: see text]. We study a new variant of Cone-TSPN where we relax this constraint and allow the cones to have non-uniform orientations. We call this problem Tilted Cone-TSPN and present a polynomial-time approximation algorithm with ratio [Formula: see text], where [Formula: see text] is the set of all cone heights. We demonstrate through simulations that our algorithm can be implemented in a practical way and that by exploiting the structure of the cones we can achieve shorter tours. Finally, we present experimental results from various agriculture applications that show the benefit of considering view angles for path planning.

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Approximation Schemes for Geometric Network Optimization Problems

Mitchell¹

2016

Encyclopedia of Algorithms

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A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics

Cited by 23 publications

References 25 publications

Approximating Watchman Routes

Approximating Watchman Routes

Approximation algorithms for tours of orientation-varying view cones

Approximation Schemes for Geometric Network Optimization Problems

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