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65
A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; t ..."
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Cited by 269 (7 self)
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In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buyatbulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
Nearoptimal hashing algorithms for approximate nearest neighbor in high dimensions
, 2008
"... In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The ..."
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Cited by 237 (4 self)
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In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The problem is of significant interest in a wide variety of areas.
Measured descent: A new embedding method for finite metrics
 In Proc. 45th FOCS
, 2004
"... We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for ..."
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Cited by 84 (26 self)
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We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any npoint metric space (X, d) embeds in Hilbert space with distortion O ( √ αX · log n), where αX is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O ( √ (log λX)log n) distortion embedding, where λX is the doubling constant of X. Since λX ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “lowdimensional, ” improved bounds are achieved. Our embeddings are volumerespecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volumerespecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted npoint planar graph O(log n) embeds in ℓ∞ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n) 2). 1
Approximation Algorithms for the Metric Labeling Problem via a New Linear Programming Formulation
, 2000
"... We consider approximation algorithms for the metric labeling problem. Informally speaking, we are given a weighted graph that specifies relations between pairs of objects drawn from a given set of objects. The goal is to find a minimum cost labeling of these objects where the cost of a labeling is d ..."
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Cited by 66 (1 self)
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We consider approximation algorithms for the metric labeling problem. Informally speaking, we are given a weighted graph that specifies relations between pairs of objects drawn from a given set of objects. The goal is to find a minimum cost labeling of these objects where the cost of a labeling is determined by the pairwise relations between the objects and a distance function on labels; the distance function is assumed to be a metric. Each object also incurs an assignment cost that is label, and vertex dependent. The problem was introduced in a recent paper by Kleinberg and Tardos [19], and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case where k is the number of labels and a 2approximation for the uniform metric case. More recently, Gupta and Tardos [14] gave a 4approximation for the truncated linear metric, a natural nonuniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a new natural integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. Specifically, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [19]. We also develop an algorithm based on our LP that achieves a ratio of 2 + p 2 ' 3:414 for the truncated linear metric improving the ratio provided by [14]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation h...
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.
A constant factor approximation for the single sink edge installation problems
 In Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing (STOC
, 2001
"... We present the first constant approximation to the single sink buyatbulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a ..."
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Cited by 63 (1 self)
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We present the first constant approximation to the single sink buyatbulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a metric. This result improves the previous bound of O(log R), where R is the set of sources. We also present a better constant approximation to the related Access Network Design problem. Our algorithms are randomized and combinatorial. As a subroutine in our algorithm, we use an interesting variant of facility location with lower bounds on the amount of demand an open facility needs to serve. We call this variant load balanced facility location, and present a constant factor approximation for it, while relaxing the lower bounds by a constant factor.
CostDistance: Two Metric Network Design
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract We present the CostDistance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of sourcesink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for CostDistance, where k is the numbe ..."
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Cited by 61 (7 self)
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Abstract We present the CostDistance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of sourcesink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for CostDistance, where k is the number of sources. We reduce many common network design problems to CostDistance, obtaining (in some cases) the first known logarithmic approximation for them. These problems include singlesink buyatbulk with variable pipe types between different sets of nodes, facility location with buyatbulk type costs on edges, and maybecast with combind cost and distance metrics. Our algorithm is also the algorithm of choice for several previous network design problems, due to its ease of implementation and fast running time. 1 Introduction Consider designing a network from the ground up. We are given a set of customers, and need to place various servers and network links in order to cheaply provide sufficient service. If we only need to place the servers, this becomes the facility location problem and constantapproximations are known. If a single server handles all customers, and we impose the additional constraint that the set of available network link types is the same for every pair of nodes (subject to constant scaling factors on cost) then this is the single sink buyatbulk problem. We give the first known approximation for the general version of this problem with both servers and network links. We reduce the network design problem to an elegant theoretical framework: the CostDistance problem. We are given a graph with a single distinguished sink node (server). Every edge in this graph can be measured along two metrics; the first will be called cost and the second will be length. Note that the two metrics are entirely independent, and that there may be any number of parallel edges in the graph. We are given a set of sources (customers). Our objective is to construct a Steiner tree connecting the sources to the sink while minimizing the combined sum of the cost of the edges in the tree and sum over sources of the weighted length from source to sink.
Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems
"... A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞ ..."
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Cited by 61 (1 self)
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A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of nonzero coefficients in a row of U, and provide a polynomialtime approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first nontrivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomialtime approximation algorithms to match this bound. This improves on the best previous mapproximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentiallymany) inequalities added. Our contribution here is to describe an appropriate
The Access Network Design Problem
 39th IEEE Symposium on Foundations of Computer Science
, 1998
"... We consider the problem of designing a minimum cost access network to carry traffic from a set of endnodes to a core network. A set of trunks of K differing types are available for leasing or buying. Some trunktypes have a high initial overhead cost but a low cost per unit bandwidth. Others have a ..."
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Cited by 47 (1 self)
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We consider the problem of designing a minimum cost access network to carry traffic from a set of endnodes to a core network. A set of trunks of K differing types are available for leasing or buying. Some trunktypes have a high initial overhead cost but a low cost per unit bandwidth. Others have a low overhead cost but a high cost per unit bandwidth. When the central core is given, we show how to construct an access network whose cost is within O(K 2 ) of optimal, under weak assumptions on the cost structure. In contrast with previous bounds, this bound is independent of the network and the traffic. Typically, the value of K is small. Our approach uses a linear programming relaxation and is motivated by a rounding technique of Shmoys, Tardos and Aardal [15]. Our techniques extend to a more complex situation in which the core is not given a priori. In this case we aim to minimize the switch cost of the core in addition to the trunk cost of the access network. We provide the same pe...