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32
Approximation algorithms for TSP with neighborhoods in the plane
 J. ALGORITHMS
, 2001
"... 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. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, incl ..."
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Cited by 90 (10 self)
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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. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, including (1) a constantfactor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearlyunit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a lineartime O(1) approximation algorithm for the case of neighborhoods that are (innite) straight lines.
On the optimal robot routing problem in wireless sensor networks
 IEEE Trans. on Knowl. and Data Eng. (TKDE
, 2007
"... Abstract—Given a set of sparsely distributed sensors in the euclidean plane, a mobile robot is required to visit all sensors to download the data and finally return to its base. The effective range of each sensor is specified by a disk, and the robot must at least reach the boundary to start communi ..."
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Cited by 29 (1 self)
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Abstract—Given a set of sparsely distributed sensors in the euclidean plane, a mobile robot is required to visit all sensors to download the data and finally return to its base. The effective range of each sensor is specified by a disk, and the robot must at least reach the boundary to start communication. The primary goal of optimization in this scenario is to minimize the traveling distance by the robot. This problem can be regarded as a special case of the Traveling Salesman Problem with Neighborhoods (TSPN), which is known to be NPhard. In this paper, we present a novel TSPN algorithm for this class of TSPN, which can yield significantly improved results compared to the latest approximation algorithm. Index Terms—Global optimization, heuristic algorithms, wireless sensor networks. Ç 1
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 25 (0 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
Approximation algorithms for Euclidean group TSP
 In Automata, languages and programming : 32nd International Colloquim, ICALP 2005
, 2005
"... Abstract. In the Euclidean group Traveling Salesman Problem (TSP), we are given a set of points P in the plane and a set of m connected regions, each containing at least one point of P. We want to find a tour of minimum length that visits at least one point in each region. This unifies the TSP with ..."
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Cited by 21 (3 self)
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Abstract. In the Euclidean group Traveling Salesman Problem (TSP), we are given a set of points P in the plane and a set of m connected regions, each containing at least one point of P. We want to find a tour of minimum length that visits at least one point in each region. This unifies the TSP with Neighborhoods and the Group Steiner Tree problem. We give a (9.1α + 1)approximation algorithm for the case when the regions are disjoint αfat objects with possibly varying size. This considerably improves the best results known, in this case, for both the group Steiner tree problem and the TSP with Neighborhoods problem. We also give the first O(1)approximation algorithm for the problem with intersecting regions. 1
NodeWeighted Steiner Tree and Group Steiner Tree in Planar Graphs
"... Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For nodeweighted Steiner tree, a classical networkoptimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We ..."
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Cited by 21 (1 self)
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Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For nodeweighted Steiner tree, a classical networkoptimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We give a constantfactor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minorclosed graph family, and also generalizes to address other optimization problems such as Steiner forest, prizecollecting Steiner tree, and networkformation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimumweight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log 3 n), or O(log 2 n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimumweight tour that must visit each group. 1
Largest and Smallest Convex Hulls for Imprecise Points
 ALGORITHMICA
, 2008
"... Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we d ..."
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Cited by 18 (4 self)
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Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NPhardness for some other variants.
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 9 (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
Approximation Algorithms for the Euclidean Traveling Salesman Problem with Discrete and Continuous Neighborhoods
, 2006
"... In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We ..."
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Cited by 7 (1 self)
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In the Euclidean traveling salesman problem with discrete neighborhoods, we are given a set of points P in the plane and a set of n connected regions (neighborhoods), each containing at least one point of P. We seek to find a tour of minimum length which visits at least one point in each region. We give (i) an O(α)approximation algorithm for the case when the regions are disjoint and αfat, with possibly varying size; (ii) an O(α³)approximation algorithm for intersecting αfat regions with comparable diameters. These results also apply to the case with continuous neighborhoods, where the sought TSP tour can hit each region at any point. We also give (iii) a simple O(log n)approximation algorithm for continuous nonfat neighborhoods. The most distinguishing features of these algorithms are their simplicity and low runningtime complexities.