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102
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 306 (8 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, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
On Approximating Arbitrary Metrics by Tree Metrics
- In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
, 1998
"... This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hi ..."
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Cited by 266 (14 self)
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This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.
Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source Buy-at-Bulk
- In Proc. of the 14 th Symposium on Discrete Algorithms (SODA
, 2003
"... We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information tha ..."
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Cited by 125 (3 self)
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We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, non-decreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buy-at-bulk network design.
Provisioning a Virtual Private Network: A network design problem for multicommodity flow
- In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must ..."
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Cited by 109 (13 self)
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Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must be connected, bandwidth in the underlying network is reserved for communication within the group, forming a virtual “sub-network.” Provisioning a virtual private network over a set of terminals gives rise to the following general network design problem. We have bounds on the cumulative amount of traffic each terminal can send and receive; we must choose a path for each pair of terminals, and a bandwidth allocation for each edge of the network, so that any traffic matrix consistent with the given upper bounds can be feasibly routed. Thus, we are seeking to design a network that can support a continuum of possible traffic scenarios. We provide optimal and approximate algorithms for several variants of this problem, depending on whether the traffic matrix is required to be symmetric, and on whether the designed network is required to be a tree (a natural constraint in a number of basic applications). We also establish a relation between this collection of network design problems and a variant of the facility location problem introduced by Karger and Minkoff; we extend their results by providing a stronger approximation algorithm for this latter problem. 1
Approximating a Finite Metric by a Small Number of Tree Metrics
- In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science
, 1998
"... Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms f ..."
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Cited by 88 (10 self)
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Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his
Lower-Stretch Spanning Trees
, 2005
"... ... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an n-vertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds i ..."
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Cited by 86 (11 self)
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... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an n-vertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the long-standing open questions in the area of low-distortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.
Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems
"... ... where all entries of c, U,and d are nonnegative. Given such a formulation, the ratio betweenthe optimal integer solution and the optimal solution to the linear program relaxation can be as bad as jjdjj1, even when U consistsof a single row. We show that by adding additional inequalities, this ra ..."
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Cited by 85 (1 self)
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... where all entries of c, U,and d are nonnegative. Given such a formulation, the ratio betweenthe optimal integer solution and the optimal solution to the linear program relaxation can be as bad as jjdjj1, even when U consistsof a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we showthat the improved ratio is bounded by the maximum number of non-zero coefficients in a row of U, and provide a polynomial-timeapproximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solutionwithin the maximum row sum times optimum. We also show that for particular instances of capacitated cov-ering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalitiesyield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. Thisis the first non-trivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical networkdesign problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we alsoprovide polynomial-time approximation algorithms to match this bound. This improves on the best previous m-approximation,where m is the number of edges in the graph. We also discuss im-provements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for thecapacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these fur-ther for outerplanar graphs. Most of our
Building Steiner trees with incomplete global knowledge
- In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... A networking problem of present day interest is that of distributing a single data item to multiple clients while minimizing network usage. Steiner tree algorithms are a natural solution method, but only when the set of clients requesting the data is known. We study what can be done without this glo ..."
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Cited by 85 (0 self)
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A networking problem of present day interest is that of distributing a single data item to multiple clients while minimizing network usage. Steiner tree algorithms are a natural solution method, but only when the set of clients requesting the data is known. We study what can be done without this global knowledge, when a given vertex knows only the probability that any other client will wish to be connected, and must simply specify a fixed path to the data to be used in case it is requested. Our problem is an example of a class of network design problems with concave cost functions (which arise when the design problem exhibits economies of scale). In order to solve our problem, we introduce a new version of the facility location problem: one in which every open facility is required to have some minimum amount of demand assigned to it. We present a simple bicriterion approximation for this problem, one which is loose in both assignment cost and minimum demand, but within a constant factor of the optimum for both. This suffices for our application. We leave open the question of finding an algorithm that produces a truly feasible approximate solution. 1.
Cost-Distance: Two Metric Network Design
- In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract We present the Cost-Distance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for Cost-Distance, where k is the numbe ..."
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Cited by 68 (7 self)
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Abstract We present the Cost-Distance problem: finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for Cost-Distance, 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 single-sink buy-at-bulk with variable pipe types between different sets of nodes, facility location with buy-at-bulk 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 constant-approximations 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 buy-at-bulk 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 Cost-Distance 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.
