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64
Guillotine subdivisions approximate polygonal subdivisions: Part II  A simple polynomialtime approximation scheme for geometric kMST, TSP, and related problems
, 1996
"... this paper, thereby achieving essentially the same results that we report here, using decomposition schemes that are somewhat similar to our own. Arora's remarkable results predate this paper by several weeks, and his discovery was done independently of this work. 2 mGuillotine Subdivisions ..."
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Cited by 165 (13 self)
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this paper, thereby achieving essentially the same results that we report here, using decomposition schemes that are somewhat similar to our own. Arora's remarkable results predate this paper by several weeks, and his discovery was done independently of this work. 2 mGuillotine Subdivisions
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
Approximation schemes for Euclidean kMedians And Related Problems
 In Proc. 30th Annu. ACM Sympos. Theory Comput
, 1998
"... In the kmedian problem we are given a set S of n points in a metric space and a positive integer k. We desire to locate k medians in space, such that the sum of the distances from each of the points of S to the nearest median is minimized. This paper gives an approximation scheme for the plane that ..."
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Cited by 121 (4 self)
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In the kmedian problem we are given a set S of n points in a metric space and a positive integer k. We desire to locate k medians in space, such that the sum of the distances from each of the points of S to the nearest median is minimized. This paper gives an approximation scheme for the plane that for any c > 0 produces a solution of cost at most 1 + 1/c times the optimum and runs in time O(n O(c+1) ). The approximation scheme also generalizes to some problems related to kmedian. Our methodology is to extend Arora's [1, 2] techniques for the TSP, which hitherto seemed inapplicable to problems such as the kmedian problem. 1 Introduction In the kmedian problem we are given a set S of n points in a metric space and a positive integer k. We desire to locate k medians in the space, such that the sum of the distances from each of the points of S to the nearest median is minimized. Besides its intrinsic appeal as a cleanlystated, basic unsolved problem in combinatorial optimizatio...
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 63 (8 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.
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...
A polynomialtime approximation scheme for weighted planar graph TSP
 PROC. 9TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS, PP 33–41
, 1998
"... Given a planar Rraph on n nodes with costs (weights) on its edges, define;he distance between nodes i &d 2 as ’ the length of the shortest path between i and i. Consider this as &I instance of me & TSP. For any E> 6, our algorithm finds a salesman tour of total cost at most (1 + E) times optimal in ..."
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Cited by 49 (13 self)
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Given a planar Rraph on n nodes with costs (weights) on its edges, define;he distance between nodes i &d 2 as ’ the length of the shortest path between i and i. Consider this as &I instance of me & TSP. For any E> 6, our algorithm finds a salesman tour of total cost at most (1 + E) times optimal in time n”(llea). We also present a quasipolynomial time algorithm for the Steiner version of this problem.
Asymptotic Theory of Greedy Approximations to Minimal KPoint Random Graphs
"... Let Xn = fx 1 ; : : : ; xn g, be an i.i.d. sample having multivariate distribution P . We derive a.s. limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples. The class includes minimal kpoint graphs constructed by the p ..."
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Cited by 42 (16 self)
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Let Xn = fx 1 ; : : : ; xn g, be an i.i.d. sample having multivariate distribution P . We derive a.s. limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples. The class includes minimal kpoint graphs constructed by the partitioning method of Ravi, Sundaram, Marathe, Rosenkrantz and Ravi [43] where the edge weight function satises the quasiadditive property of Redmond and Yukich [45]. In particular this includes greedy approximations to the kpoint minimal spanning tree (kMST), Steiner tree (kST), and the traveling salesman problem (kTSP). An expression for the inuence function of the minimal weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution. The inuence function takes a form which indicates that the kpoint minimal graph in d > 1 dimensions has robustness properties in IR d which are analogous to those of rank order statistics in one dimension. A direct result of our theory is that the logweight of the kpoint minimal graph is a consistent nonparametric estimate of the Renyi entropy of the distribution P . Possible applications of this work include: analysis of random communication network topologies, estimation of the mixing coecient in contaminated mixture models, outlier discrimination and rejection, clustering and pattern recognition, robust nonparametric regression, two sample matching and image registration.
On Traveling Salesperson Problems for Dubins' vehicle: stochastic and dynamic environments
 CDC 2005, TO APPEAR
, 2005
"... In this paper we propose some novel planning and routing strategies for Dubins’ vehicle, i.e., for a nonholonomic vehicle moving along paths with bounded curvature, without reversing direction. First, we study a stochastic version of the Traveling Salesperson Problem (TSP): given n targets randomly ..."
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Cited by 42 (13 self)
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In this paper we propose some novel planning and routing strategies for Dubins’ vehicle, i.e., for a nonholonomic vehicle moving along paths with bounded curvature, without reversing direction. First, we study a stochastic version of the Traveling Salesperson Problem (TSP): given n targets randomly sampled from a uniform distribution in a rectangle, what is the shortest Dubins ’ tour through the targets and what is its length? We show that the expected length of such a tour is Ω(n 2/3) and we propose a novel algorithm that generates a tour of length O(n 2/3 log(n) 1/3) with high probability. Second, we study a dynamic version of the TSP (known as “Dynamic Traveling Repairperson Problem” in the Operations Research literature): given a stochastic process that generates targets, is there a policy that allows a Dubins vehicle to stabilize the system, in the sense that the number of unvisited targets does not diverge over time? If such policies exist, what is the minimum expected waiting period between the time a target is generated and the time it is visited? We propose a novel recedinghorizon algorithm whose performance is almost within a constant factor from the optimum.
Parameterized Complexity for the Skeptic
 In Proc. 18th IEEE Annual Conference on Computational Complexity
, 2003
"... The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity. ..."
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Cited by 36 (1 self)
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The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
On the efficiency of polynomial time approximation schemes
, 1997
"... A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind t ..."
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Cited by 27 (0 self)
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A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new, and sometimes harder, techniques. For some interesting problems, only n”(“E) approximation schemes are known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time f ( l/n)nO(‘), no matter how fast function f grows. Our result relies on a connection with Parameterized Complexity Theory, and we show that this connection is necessary.