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50
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
 In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96
, 1996
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes a ..."
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Cited by 320 (3 self)
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Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
Bypassing the embedding: Algorithms for lowdimensional metrics
 In Proceedings of the 36th ACM Symposium on the Theory of Computing (STOC
, 2004
"... The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2 k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into l ..."
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Cited by 65 (4 self)
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The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2 k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into low dimensional Euclidean spaces, they would inherit several algorithmic and structural properties of the Euclidean spaces. Unfortunately however, such a restriction on dimension does not suffice to guarantee embeddibility in a normed space. In this paper we explore the option of bypassing the embedding. In particular we show the following for low dimensional metrics: • Quasipolynomial time (1+ɛ)approximation algorithm for various optimization problems such as TSP, kmedian and facility location. • (1 + ɛ)approximate distance labeling scheme with optimal label length. • (1+ɛ)stretch polylogarithmic storage routing scheme.
Approximating the stretch factor of Euclidean paths, cycles and trees
 SIAM J. Comput
, 1999
"... Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and ..."
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Cited by 48 (7 self)
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Given a set S of n points in R d , and a graph G having the points of S as its vertices, the stretch factor t of G is dened as the maximal value jpqj G =jpqj, where p; q 2 S, p 6= q, jpqj G is the length of a shortest path in G between p and q, and jpqj is the Euclidean distance between p and q. We consider the problem of designing algorithms that, for an arbitrary constant > 0, compute an approximation to this stretch factor, i.e., a value t such that t t (1 + )t. We give eÆcient solutions for the cases when G is a path, cycle, or tree. The main idea used in all the algorithms is to use wellseparated pair decompositions to speed up the computations. 1 Introduction Let S be a set of n points in R d , where d 1 is a small constant, and let G be an undirected connected graph having the points of S as its vertices. The length of any edge (p; q) of G is dened as the Euclidean distance jpqj between the two vertices p and q. The length of a path in G is dened a...
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 42 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
A polynomialtime approximation scheme for Steiner tree in planar graphs
 In Proceedings of the 18th Annual ACMSIAM Symposium on Discrete Algorithms
, 2007
"... We give an O(n log n) approximation scheme for Steiner tree in planar graphs. 1 ..."
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Cited by 31 (13 self)
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We give an O(n log n) approximation scheme for Steiner tree in planar graphs. 1
Steiner tree in planar graphs: An O(n log n) approximation scheme with singly exponential dependence on epsilon
 Proc. 10th Ann. Workshop on Algorithms and Data Structures
, 2007
"... Abstract. We give an algorithm that, for any ɛ>0, any undirected planar graph G,andanysetS of nodes of G, computes a (1+ɛ)optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2 poly(1/ɛ) n log n). 1 ..."
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Cited by 16 (10 self)
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Abstract. We give an algorithm that, for any ɛ>0, any undirected planar graph G,andanysetS of nodes of G, computes a (1+ɛ)optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2 poly(1/ɛ) n log n). 1
Approximating a Minimum Manhattan Network
 Nordic J. Comput
, 1999
"... Given a set S of n points in the plane, we dene a Manhattan Network on S as a rectilinear network G with the property that for every pair of points in S, the network G contains the shortest rectilinear path between them. A Minimum Manhattan Network on S is a Manhattan network of minimum possible ..."
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Cited by 15 (1 self)
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Given a set S of n points in the plane, we dene a Manhattan Network on S as a rectilinear network G with the property that for every pair of points in S, the network G contains the shortest rectilinear path between them. A Minimum Manhattan Network on S is a Manhattan network of minimum possible length. A Manhattan network can be thought of as a graph G = (V; E), where the vertex set V corresponds to points from S and a set of Steiner points S 0 , and the edges in E correspond to horizontal or vertical line segments connecting points in S [S 0 .
FaultTolerant Geometric Spanners
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2004
"... We present two new results about vertex and edge faulttolerant spanners in Euclidean spaces. We describe the first construction of vertex and edge faulttolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t> 1 and any nonnegative in ..."
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Cited by 15 (1 self)
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We present two new results about vertex and edge faulttolerant spanners in Euclidean spaces. We describe the first construction of vertex and edge faulttolerant spanners having optimal bounds for maximum degree and total cost. We present a greedy algorithm that for any t> 1 and any nonnegative integer k, constructs a kfaulttolerant tspanner in which every vertex is of degree O(k) and whose total cost is O(k2) times the cost of the minimum spanning tree; these bounds are asymptotically optimal. Our next contribution is an efficient algorithm for constructing good faulttolerant spanners. We present a new, sufficient condition for a graph to be a kfaulttolerant spanner. Using this condition, we design an efficient algorithm that finds faulttolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost.
The traveling salesman problem with few inner points
 In Proc. 10th COCOON, volume 3106 of LNCS
, 2004
"... We propose two algorithms for the planar Euclidean traveling salesman problem. The first runs in O(k!kn) time and O(k) space, and the second runs in O(2 k k 2 n) time and O(2 k kn) space, where n denotes the number of input points and k denotes the number of points interior to the convex hull. ..."
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Cited by 12 (2 self)
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We propose two algorithms for the planar Euclidean traveling salesman problem. The first runs in O(k!kn) time and O(k) space, and the second runs in O(2 k k 2 n) time and O(2 k kn) space, where n denotes the number of input points and k denotes the number of points interior to the convex hull.
Fast approximation schemes for Euclidean multiconnectivity problems (Extended Abstract)
 PROC. 27TH ANNUAL INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP
, 2000
"... We present new polynomialtime approximation schemes (PTAS) for several basic minimumcost multiconnectivity problems in geometrical graphs. We focus on low connectivity requirements. Each of our schemes either significantly improves the previously known upper timebound or is the first PTAS for t ..."
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Cited by 11 (5 self)
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We present new polynomialtime approximation schemes (PTAS) for several basic minimumcost multiconnectivity problems in geometrical graphs. We focus on low connectivity requirements. Each of our schemes either significantly improves the previously known upper timebound or is the first PTAS for the considered problem. We provide a randomized approximation scheme for finding a biconnected graph spanning a set of points in a multidimensional Euclidean space and having the expected total cost within (1+ε) of the optimum. For any constant dimension and ε, our scheme runs in time O(n log n). It can be turned into Las Vegas one without affecting its asymptotic time complexity, and also efficiently derandomized. The only previously known truly polynomialtime approximation (randomized) scheme for this problem runs in expected time n · (log n) O((log log n)9) in the simplest planar case. The efficiency of our scheme relies on transformations of nearly optimal low cost special spanners into submultigraphs having good decomposition