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A constantfactor approximation algorithm for the k mst problem (extended abstract
 in Proceedings of the twentyeighth annual ACM symposium on Theory of computing (STOC ’96
, 1996
"... ABSTRACT In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constantfactor approximation algorithm for planar TSPN with pairwisedisjoint connecte ..."
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Cited by 58 (5 self)
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ABSTRACT In the Euclidean TSP with neighborhoods (TSPN) problem we seek a shortest tour that visits a given set of n neighborhoods. The Euclidean TSPN generalizes the standard TSP on points. We present the first constantfactor approximation algorithm for planar TSPN with pairwisedisjoint connected neighborhoods of any size or shape. Prior approximation bounds were O(log n), except in special cases. The methods also apply to the case of arbitrarily overlapping regions that are convex.
A PTAS for TSP with neighborhoods among fat regions in the plane
 In Proc. ACMSIAM SODA’07
, 2007
"... The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of n regions (neighborhoods). We present the first polynomialtime approximation scheme for TSPN for a set of regions given by arbitrary disjoint fat regions in the plane. This improves substan ..."
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Cited by 26 (1 self)
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The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of n regions (neighborhoods). We present the first polynomialtime 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 noncomparable sizes. Our result is based on a novel extension of the mguillotine method. The result applies to regions that are “fat ” in a very weak sense: each region Pi has area Ω([diam(Pi)] 2), but is otherwise arbitrary. 1
A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics
"... We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of ..."
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Cited by 8 (0 self)
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We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of
The traveling salesman problem for lines, balls and planes
, 2013
"... We revisit the traveling salesman problem with neighborhoods (TSPN) and obtain several approximation algorithms. These constitute either improvements over previously best approximations achievable in comparable times (for unit disks in the plane), or first approximations ever (for hyperplanes and li ..."
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Cited by 2 (1 self)
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We revisit the traveling salesman problem with neighborhoods (TSPN) and obtain several approximation algorithms. These constitute either improvements over previously best approximations achievable in comparable times (for unit disks in the plane), or first approximations ever (for hyperplanes and lines in R d, and unit balls in R 3). (I) Given a set of n hyperplanes in R d, a TSP tour that is at most O(1) times longer than the optimal can be computed in O(n) time, when d is constant. (II) Given a set of n lines in R d, a TSP tour that is at most O(log 3 n) times longer than the optimal can be computed in polynomial time, when d is constant. (III) Given a set of n unit disks in the plane or n unit balls in R 3, we improve the approximation ratios relying on a black box that computes a good approximate tour for a set of points in the ambient space (in our case, these are the centers of a subset of the disks or the balls).
ConstantFactor Approximation for TSP with Disks
, 2015
"... We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constantratio approximation for disks in the plane: Given a set of n disks in the plane, a TSP tour whose length is at most O(1) times the optimal with high probability can be computed in time that is polynomi ..."
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We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constantratio approximation for disks in the plane: Given a set of n disks in the plane, a TSP tour whose length is at most O(1) times the optimal with high probability can be computed in time that is polynomial in n. Our result is the first constantratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a O(1)approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.