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84
Learning to Order Things
 Journal of Artificial Intelligence Research
, 1998
"... There are many applications in which it is desirable to order rather than classify instances. Here we consider the problem of learning how to order, given feedback in the form of preference judgments, i.e., statements to the effect that one instance should be ranked ahead of another. We outline a ..."
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Cited by 409 (12 self)
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There are many applications in which it is desirable to order rather than classify instances. Here we consider the problem of learning how to order, given feedback in the form of preference judgments, i.e., statements to the effect that one instance should be ranked ahead of another. We outline a twostage approach in which one first learns by conventional means a preference function, of the form PREF(u; v), which indicates whether it is advisable to rank u before v. New instances are then ordered so as to maximize agreements with the learned preference function. We show that the problem of finding the ordering that agrees best with a preference function is NPcomplete, even under very restrictive assumptions. Nevertheless, we describe a simple greedy algorithm that is guaranteed to find a good approximation. We then discuss an online learning algorithm, based on the "Hedge" algorithm, for finding a good linear combination of ranking "experts." We use the ordering algorith...
Multicommodity maxflow mincut theorems and their use in designing approximation algorithms
 J. ACM
, 1999
"... Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound imp ..."
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Cited by 370 (6 self)
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Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by the mincut. The result (which is existentially optimal) establishes an important analogue of the famous 1commodity maxflow mincut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomialtime (polylog ntimesoptimal) approximation algorithms for wellknown NPhard optimization problems such as graph partitioning, mincut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
Euclidean distortion and the Sparsest Cut
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 120 (25 self)
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BiLipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
Approximating Fractional Multicommodity Flow Independent of the Number of Commodities
, 1999
"... We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the ru ..."
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Cited by 114 (8 self)
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We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, performing in O (ffl \Gamma2 m 2 ) time. For maximum concurrent flow, and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k ? m=n. Our algorithms build on the framework proposed by Garg and Konemann [4]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k= log W where W is ...
A Polylogarithmic Approximation of the Minimum Bisection
, 2001
"... A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets. ..."
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Cited by 89 (7 self)
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A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets.
Proving Integrality Gaps Without Knowing the Linear Program
 Theory of Computing
, 2002
"... Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods see ..."
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Cited by 63 (2 self)
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Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.
Embeddings of negativetype metrics and an improved approximation to generalized sparsest cut
, 2007
"... In this paper, we study metrics of negative type, which are metrics (V, d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding resu ..."
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Cited by 52 (0 self)
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In this paper, we study metrics of negative type, which are metrics (V, d) such that √ d is an Euclidean metric; these metrics are thus also known as “ℓ2squared” metrics. We show how to embed npoint negativetype metrics into Euclidean space ℓ2 with distortion D = O(log 3/4 n). This embedding result, in turn, implies an O(log 3/4 k)approximation algorithm for the Sparsest Cut problem with nonuniform demands. Another corollary we obtain is that npoint subsets of ℓ1 embed into ℓ2 with distortion O(log 3/4 n).
Cuts, trees and l1embeddings of graphs
, 2002
"... Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ` 1 space. The main results are: 1. Explicit constantdistortion embeddings of all s ..."
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Cited by 31 (3 self)
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Motivated by many recent algorithmic applications, this paper aims to promote a systematic study of the relationship between the topology of a graph and the metric distortion incurred when the graph is embedded into ` 1 space. The main results are: 1. Explicit constantdistortion embeddings of all seriesparallel graphs, and all graphs with bounded Euler number. These are the rst natural families known to have constant distortion (strictly greater than 1). Using the above embeddings, algorithms are obtained which approximate the sparsest cut in such graphs to within a constant factor. 2. A constantdistortion embedding of outerplanar graphs into the restricted class of ` 1 metrics known as \dominating tree metrics". A lower bound of (logn) on the distortion for embeddings of seriesparallel graphs into (distributions over) dominating tree metrics
Faster Approximation Schemes for Fractional Multicommodity Flow Problems
"... We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K"onemann [7] and Fleischer [5] we can reduce their running ..."
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Cited by 28 (0 self)
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We present fully polynomial approximation schemes for concurrent multicommodity flow problems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K&quot;onemann [7] and Fleischer [5] we can reduce their running time on a graph with n vertices and m edges from ~O(&quot;2(m2 + km)) to ~O(&quot;2m2) foran implicit representation of the output, or ~ O(&quot;2(m2 + kn)) for an explicit representation, where ~ O(f) denotes a quantity that is O(f logO(1) m). The implicit representation consists of a set oftrees rooted at sources (there can be more than one tree per source), and with sinks as their leaves, together with flow values for the flow directed from the source to the sinks in a particular tree.Given this implicit representation, the approximate value of the concurrent flow is known, but if we want the explicit flow per commodity per edge, we would have to combine all these trees together,and the cost of doing so may be prohibitive. In case we want to calculate explicitly the solution flow, we modify our schemes so that they run in time polylogarithmic in nk (n is the numberof nodes in the network). This is within a polylogarithmic factor of the trivial lower bound of time \Omega (nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a polylogarithmic factor of the minimum possible dependency of the running time on the number of commodities k.