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20
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 19 (4 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 10 (0 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 10 (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.
Approximation Schemes for the Generalized Geometric Problems with Geographic Clustering
 EWCG 2005
, 2005
"... This paper is concerned with polynomial time approximations schemes for the generalized geometric problems with geographic clustering. We illustrate the approach on the generalized traveling salesman problem which is also known as GroupTSP or TSP with neighborhoods. We prove that under the conditio ..."
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Cited by 6 (1 self)
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This paper is concerned with polynomial time approximations schemes for the generalized geometric problems with geographic clustering. We illustrate the approach on the generalized traveling salesman problem which is also known as GroupTSP or TSP with neighborhoods. We prove that under the condition that all regions are nonintersecting and have comparable sizes and shapes, the problem admits PTAS. To derive a PTAS we extend the algorithm by Arora [2]. This extension involves the dissection mechanism and solution of the selection problem. We observe that the results are applicable to many generalized geometric problems, to other Minkowski norms, and to other fixed dimensional spaces.
Combinatorial optimization with explicit delineation of the ground set by a collection of subsets
 In SIAM J. Discrete Math
, 2008
"... We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund, 1997) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a part ..."
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Cited by 5 (0 self)
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We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund, 1997) given that the ground set of elements N has additional characteristics. For each problem in this paper, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well known classical graph problems and include the minimal spanning tree, the assignment problem, a number of elementary machine scheduling problems, binpacking, and the TSP. We conclude that for all these generalized problems the existence of PTAS (polynomial time approximation scheme) is impossible unless P=NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality two.
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 4 (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 3 (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.
Existence of simple tours of imprecise points
 IN PROC. 23RD ANNU. EUROPEAN WORKSHOP ON COMPUTATIONAL GEOMETRY
, 2007
"... Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NPcomplete to decide whether it is possible to place the points inside their regions in such a way that the res ..."
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Cited by 3 (0 self)
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Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NPcomplete to decide whether it is possible to place the points inside their regions in such a way that the resulting polygon is simple. Furthermore, it is NPhard to minimize the length of a simple tour visiting the regions in order, when the connections between consecutive regions do not need to be straight line segments.
Approximation Algorithms for the MinimumLength Corridor and Related Problems
, 2007
"... Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the bou ..."
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Cited by 3 (2 self)
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Given a rectangular boundary partitioned into rectangles, the MinimumLength Corridor (MLCR) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the boundary of the rectangles, and must include at least one point from the boundary of every rectangle and from the rectangular boundary. The MLCR problem has been shown to be NPhard. In this paper we present the first polynomial time constant ratio approximation algorithm for the MLCR and MLCn problems. The MLCn problem is a generalization of the the MLCR problem where the rectangles are rectilinear kgons, for k ≤ n. We also present a polynomial time constant ratio approximation algorithm for the Group Traveling Salesperson Problem (GTSP) for a rectangle partitioned into rectilinear kgons as in the MLCn problem.