Article

A PTAS for TSP with neighborhoods among fat regions in the plane

Author:

Joseph S. B. MitchellAuthors Info & Claims

SODA '07: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms

Pages 11 - 18

Published: 07 January 2007 Publication History

Abstract

The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of n regions (neighborhoods). We present the first polynomial-time approximation scheme for TSPN for a set of regions given by arbitrary disjoint fat regions in the plane. This improves substantially upon the known approximation algorithms, and is the first PTAS for TSPN on regions of non-comparable sizes. Our result is based on a novel extension of the m-guillotine method. The result applies to regions that are "fat" in a very weak sense: each region P_i contains a disk of radius Ω(diam(P_i)), but is otherwise arbitrary. Further, the result applies even if the regions intersect arbitrarily, provided that there exists a packing of disjoint disks, of radii Ω(diam(P_i)), contained within their respective regions. Finally, the PTAS result applies also to the case in which the regions are sets of points or polygons, each each lying within one of a given set of disjoint fat regions.

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Antoniadis AFleszar KHoeksma RSchewior KChan T(2019)A PTAS for euclidean TSP with hyperplane neighborhoodsProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310502(1089-1105)Online publication date: 6-Jan-2019
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A PTAS for TSP with neighborhoods among fat regions in the plane
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
  2. Mathematical analysis
    1. Functional analysis
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Conferences

SODA '07: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms

January 2007

1322 pages

ISBN:9780898716245

Conference Chair:
Harold Gabow
University of Colorado, Boulder

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 07 January 2007

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SODA '07 Paper Acceptance Rate 139 of 382 submissions, 36%;

Overall Acceptance Rate 411 of 1,322 submissions, 31%

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Cited By

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https://dl.acm.org/doi/10.1177/0278364919893455
Antoniadis AFleszar KHoeksma RSchewior KChan T(2019)A PTAS for euclidean TSP with hyperplane neighborhoodsProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310502(1089-1105)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310502
Plonski PIsler V(2019)Approximation algorithms for tours of height-varying view conesInternational Journal of Robotics Research10.1177/027836491878435338:2-3(224-235)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1177/0278364918784353
Chan TJiang S(2018)Reducing Curse of DimensionalityACM Transactions on Algorithms10.1145/315823214:1(1-18)Online publication date: 3-Jan-2018
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Chambers EErickson AFekete SLenchner JSember JSrinivasan VStege UStolpner SWeibel CWhitesides S(2017)Connectivity Graphs of Uncertainty RegionsAlgorithmica10.5555/3125403.312541878:3(990-1019)Online publication date: 1-Jul-2017
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Chan TJiang S(2016)Reducing curse of dimensionalityProceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms10.5555/2884435.2884489(754-765)Online publication date: 10-Jan-2016
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Hajiaghayi MKhandekar RKhani MKortsarz G(2016)Approximation Algorithms for Movement RepairmenACM Transactions on Algorithms10.1145/290873712:4(1-38)Online publication date: 3-Aug-2016
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Bateni MDemaine EHajiaghayi MMarx DWichs DMansour Y(2016)A PTAS for planar group Steiner tree via spanner bootstrapping and prize collectingProceedings of the forty-eighth annual ACM symposium on Theory of Computing10.1145/2897518.2897549(570-583)Online publication date: 19-Jun-2016
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Dumitrescu ATóth C(2016)The Traveling Salesman Problem for Lines, Balls, and PlanesACM Transactions on Algorithms10.1145/285041812:3(1-29)Online publication date: 25-Apr-2016
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