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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 147 (12 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Some NPcomplete Geometric Problems
"... We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NPhard i ..."
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Cited by 83 (2 self)
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We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NPhard if the distance I~asure is the (unmodified) Euclidean metric. However, for reasons we discuss, there is some question as to whether these problems, or even the wellsolved MINIMUM SPANNING TREE problem, are in NP when the distance measure is the Euclidean metric.
Multicast Routing with EndtoEnd Delay and Delay Variation Constraints
 IEEE Journal on Selected Areas in Communications
, 1995
"... We study the problem of constructing multicast trees to meet the quality of service requirements of realtime, interactive applications operating in highspeed packetswitched environments. In particular, we assume that multicast communication depends on (a) bounded delay along the paths from the so ..."
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Cited by 70 (2 self)
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We study the problem of constructing multicast trees to meet the quality of service requirements of realtime, interactive applications operating in highspeed packetswitched environments. In particular, we assume that multicast communication depends on (a) bounded delay along the paths from the source to each destination, and (b) bounded variation among the delays along these paths. We first establish that the problem of determining such a constrained tree is NPcomplete. We then derive heuristics that demonstrate good average case behavior in terms of the maximum interdestination delay variation of the final tree. In addition, our heuristics achieve their best performance under conditions typical of multicast scenarios in highspeed networks. We also show that it is possible to dynamically reorganize the initial tree in response to changes in the destination set, in a way that is minimally disruptive to the multicast session. Department of Computer Science North Carolina State Uni...
Approximating Geometrical Graphs Via Spanners and Banyans
, 1998
"... The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. ..."
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Cited by 61 (0 self)
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The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. An interesting byproduct of our work is the definition and construction of banyans, a generalization of graph spanners. A (1 + ffl)banyan for a set of points A is a set of points A 0 and line segments S with endpoints in A [ A 0 such that a 1 + ffl optimal Steiner Minimum Tree for any subset of A is contained in S. We give a construction for banyans such that the total length of the line segments in S is within a constant factor of the length of the minimum spanning tree of A, and jA 0 j = O(jAj), when ffl and d are fixed. In this abbreviated paper, we only provide proofs of these results in two dimensions. The full paper on WDS's web page (http://www.neci.nj.nec.com/homepages/wds, c...
When Hamming Meets Euclid: The Approximability of Geometric TSP and MST (Extended Abstract)
, 1997
"... We prove that the Traveling Salesperson Problem (MIN TSP) and the Minimum Steiner Tree Problem (MIN ST) are Max SNPhard (and thus NPhard to approximate within some constant r ? 1) even if all cities (respectively, points) lie in the geometric space R n (n is the number of cities/points) and ..."
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Cited by 41 (2 self)
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We prove that the Traveling Salesperson Problem (MIN TSP) and the Minimum Steiner Tree Problem (MIN ST) are Max SNPhard (and thus NPhard to approximate within some constant r ? 1) even if all cities (respectively, points) lie in the geometric space R n (n is the number of cities/points) and distances are computed with respect to the l 1 (rectilinear) metric. The TSP hardness results also hold for any l p metric, including the Euclidean metric, and in R logn . The running time of Arora's approximation scheme for geometric MIN TSP in R d is doubly exponential in d. Our results imply that this dependance is necessary unless NP has subexponential algorithms. We also prove, as an intermediate step, the hardness of approximating MIN TSP and MIN ST in Hamming spaces. The reduction for MIN TSP uses errorcorrecting codes and random sampling; the reduction for MIN ST uses the integrality property of MINCUT. The only previous nonapproximability results for ...
On the complexity of approximating TSP with neighborhoods and related problems
 Computational Complexity
"... We prove that various geometric covering problems, related to the Travelling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the GroupTravelling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the GroupSteinerTree in ..."
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Cited by 30 (2 self)
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We prove that various geometric covering problems, related to the Travelling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the GroupTravelling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the GroupSteinerTree in the Euclidean plane and the Minimum Watchman Tour and the Minimum Watchman Path in 3D. It resolves three open problems presented in the comprehensive survey of Mitchell [Mit00], improves a previously known approximation hardness factor of 2041 2040 [GL00, dBGK+ 02] for the first problem, and it is the first approximation hardness factor for the other problems. Some inapproximability factors are also shown for special cases of the above problems, where the size of the sets is bounded. GroupTSP and GroupSteinerTree where each neighbourhood is connected are also considered. It is shown that approximating these variants to within any constant factor smaller than 2, is NPhard. For the GroupTravelling Salesman and GroupSteinerTree Problems in dimension d, we show an innapproximability factor of O(log dâˆ’1 d HyperGraph VertexCover.
ResourceConstrained Geometric Network Optimization (Extended Abstract)
"... We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wel ..."
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Cited by 21 (1 self)
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We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the wellpublicized open problem on the approximability of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2approximation for this problem. We also obtain approximation algorithms for variants of this problem in which the network required is a tree (3approximation) or a path (2approximation). No prior approximation bounds were known for any of these problems. We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2approximation for both the cycle and tree versions of the problem on points in the plane, as well as a ...
The Rectilinear Steiner Arborescence Problem is NPComplete
, 2000
"... Given a set P of points in the first quadrant, a Rectilinear Steiner Arborescence (RSA) is a directed tree rooted at the origin, containing all points in P , and composed solely of horizontal and vertical edges oriented from left to right, or from bottom to top. The complexity of finding an RSA with ..."
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Cited by 21 (0 self)
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Given a set P of points in the first quadrant, a Rectilinear Steiner Arborescence (RSA) is a directed tree rooted at the origin, containing all points in P , and composed solely of horizontal and vertical edges oriented from left to right, or from bottom to top. The complexity of finding an RSA with the minimum total edge length for general planar point sets has been a major open problem, and has important applications in VLSI. In this paper, we prove the problem is strongly NPcomplete. The proof also shows the Euclidean version of the Steiner Arborescence problem is NPhard.