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35
The kserver problem
 Computer Science Review
"... The kserver problem is perhaps the most influential online problem: natural, crisp, with a surprising technical depth that manifests the richness of competitive analysis. The kserver conjecture, which was posed more that two decades ago when the problem was first studied within the competitive ana ..."
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Cited by 66 (5 self)
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The kserver problem is perhaps the most influential online problem: natural, crisp, with a surprising technical depth that manifests the richness of competitive analysis. The kserver conjecture, which was posed more that two decades ago when the problem was first studied within the competitive analysis framework, is still open and has been a major driving force for the development of the area online algorithms. This article surveys some major results for the kserver. 1
LowerStretch 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 nvertex 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 63 (10 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 nvertex 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 longstanding open questions in the area of lowdistortion 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 lowstretch spanning trees.
Embeddings of negativetype metrics and an improved approximation to generalized sparsest cut
 IN SODA ‘05: PROCEEDINGS OF THE SIXTEENTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2005
"... In this paper, we study the 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 re ..."
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Cited by 43 (0 self)
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In this paper, we study the 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).
A general approach to online network optimization problems
 ACM Transactions on Algorithms
, 2004
"... ..."
An Efficient CostSharing Mechanism for the PrizeCollecting Steiner Forest Problem
"... In an instance of the prizecollecting Steiner forest problem (PCSF) we are given an undirected graph G = (V,E), nonnegative edgecosts c(e) for all e ∈ E, terminal pairs R = {(si,ti)}1≤i≤k, and penalties π1,...,πk. A feasible solution (F,Q) consists of a forest F and a subset Q of terminal pairs s ..."
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Cited by 18 (4 self)
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In an instance of the prizecollecting Steiner forest problem (PCSF) we are given an undirected graph G = (V,E), nonnegative edgecosts c(e) for all e ∈ E, terminal pairs R = {(si,ti)}1≤i≤k, and penalties π1,...,πk. A feasible solution (F,Q) consists of a forest F and a subset Q of terminal pairs such that for all (si,ti) ∈ R either si,ti are connected by F or (si,ti) ∈ Q. The objective is to compute a feasible solution of minimum cost c(F) + π(Q). A gametheoretic version of the above problem has k players, one for each terminalpair in R. Player i’s ultimate goal is to connect si and ti, and the player derives a privately held utility ui ≥ 0 from being connected. A service provider can connect the terminals si and ti of player i in two ways: (1) by buying the edges of an si,tipath in G, or (2) by buying an alternate connection between si and ti (maybe from some other provider) at a cost of πi. In this paper, we present a simple 3budgetbalanced and groupstrategyproof mechanism for the above problem. We also show that our mechanism computes client sets whose social cost is at most O(log 2 k) times the minimum social cost of any player set. This matches a lowerbound that was recently given by Roughgarden and Sundararajan (STOC ’06).
A SketchBased Distance Oracle for WebScale Graphs
"... We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves t ..."
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Cited by 16 (1 self)
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We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the timeconsuming shortestpath computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small “sketch ” for each node in the graph, and at querytime we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance. Categories and Subject Descriptors G.2.2 [Graph Theory]: Graph algorithms, path and circuit problems
Online and Stochastic Survivable Network Design
"... Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is kno ..."
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Cited by 11 (3 self)
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Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edgeconnectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict costshares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rentorbuy version and (b) the (twostage) stochastic version of the edgeconnected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edgecosts are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)strict cost shares, which gives constantfactor rentorbuy and stochastic algorithms for these instances.
The polymatroid Steiner problems
 J. Comb. Optim
, 2005
"... Abstract. The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S ⊆ V in a weighted graph G = (V, E, c), c: E → R +. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P (V) is de ..."
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Cited by 9 (0 self)
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Abstract. The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S ⊆ V in a weighted graph G = (V, E, c), c: E → R +. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P (V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S ⊆ V). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s1,..., sk}, each sensor si can choose to monitor only a single target from a subset of targets Xi, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X1 ∪... ∪ Xk. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et. al. [7]. We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Charikar et al [6].
Differentially Private Approximation Algorithms
 In Proceedings of the ACMSIAM Symposium on Discrete Algorithms
, 2010
"... Consider the following problem: given a metric space, some of whose points are “clients, ” select a set of at most k facility locations to minimize the average distance from the clients to their nearest facility. This is just the wellstudied kmedian problem, for which many approximation algorithms ..."
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Cited by 8 (3 self)
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Consider the following problem: given a metric space, some of whose points are “clients, ” select a set of at most k facility locations to minimize the average distance from the clients to their nearest facility. This is just the wellstudied kmedian problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. This raises the following quandary: what if the locations of the clients are sensitive information that we would like to keep private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients? In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any nontrivial approximation to even the value of an optimal solution, let alone the entire solution. Apart from the kmedian problem, we consider the problems of vertex and set cover, mincut, kmedian,
Spanners with slack
 Proceedings of the 14th European symposium on algorithms
, 2006
"... Abstract. Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily l ..."
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Cited by 7 (1 self)
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Abstract. Given a metric (V,d), a spanner is a sparse graph whose shortestpath metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distances, but are then required to incur only a constant (independent of n) distortion on the remaining distances? We answer this question in the affirmative, thus complementing similar recent results on embeddings with slack into ℓp spaces. For instance, we show that if we ignore an ɛ fraction of the distances, we can get spanners with O(n) edgesand O(log 1) distortion for the remaining distances. ɛ We also show how to obtain sparse and lowweight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces. 1