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206
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 238 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
Characterizing the capacity region in multiradio multichannel wireless mesh networks
 in ACM MobiCom
, 2005
"... Next generation fixed wireless broadband networks are being increasingly deployed as mesh networks in order to provide and extend access to the internet. These networks are characterized by the use of multiple orthogonal channels and nodes with the ability to simultaneously communicate with many nei ..."
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Cited by 146 (0 self)
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Next generation fixed wireless broadband networks are being increasingly deployed as mesh networks in order to provide and extend access to the internet. These networks are characterized by the use of multiple orthogonal channels and nodes with the ability to simultaneously communicate with many neighbors using multiple radios (interfaces) over orthogonal channels. Networks based on the IEEE 802.11a/b/g and 802.16 standards are examples of these systems. However, due to the limited number of available orthogonal channels, interference is still a factor in such networks. In this paper, we propose a network model that captures the key practical aspects of such systems and characterize the constraints binding their behavior. We provide necessary conditions to verify the feasibility of rate vectors in these networks, and use them to derive upper bounds on the capacity in terms of achievable throughput, using a fast primaldual algorithm. We then develop two link channel assignment schemes, one static and the other dynamic, in order to derive lower bounds on the achievable throughput. We demonstrate through simulations that the dynamic link channel assignment scheme performs close to optimal on the average, while the static link channel assignment algorithm also performs very well. The methods proposed in this paper can be a valuable tool for network designers in planning network deployment and for optimizing different performance objectives.
The Complexity of Pure Nash Equilibria
, 2004
"... We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. ..."
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Cited by 145 (6 self)
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We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. We discuss implications to nonatomic congestion games, and we explore the scope of the potential function method for proving existence of pure Nash equilibria.
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 96 (6 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 ...
Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice
, 2001
"... After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming ..."
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Cited by 69 (4 self)
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After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.
A Combinatorial, PrimalDual approach to Semidefinite Programs
"... Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced S ..."
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Cited by 65 (11 self)
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Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced Separator in undirected and directed weighted graphs, and the Min UnCut problem, this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primaldual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones.
Approximation techniques for utilitarian mechanism design
 IN PROC. 36TH ACM SYMP. ON THEORY OF COMPUTING
, 2005
"... This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multiparameter agents. We focus on approximation algorithms for NPhard mechanism design problems. These algorithms need to satisfy certain monotonic ..."
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Cited by 64 (3 self)
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This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multiparameter agents. We focus on approximation algorithms for NPhard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for singleminded multiunit auctions. The best previous result for such auctions was a 2approximation. In addition,
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
Crosslayer optimization in TCP/IP networks
 IEEE/ACM Transactions on Networking
, 2005
"... Abstract — TCP–AQM can be interpreted as distributed primaldual algorithms to maximize aggregate utility over source rates. We show that an equilibrium of TCP/IP, if exists, maximizes aggregate utility over both source rates and routes, provided congestion prices are used as link costs. An equilibr ..."
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Cited by 53 (8 self)
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Abstract — TCP–AQM can be interpreted as distributed primaldual algorithms to maximize aggregate utility over source rates. We show that an equilibrium of TCP/IP, if exists, maximizes aggregate utility over both source rates and routes, provided congestion prices are used as link costs. An equilibrium exists if and only if this utility maximization problem and its Lagrangian dual have no duality gap. In this case, TCP/IP incurs no penalty in not splitting traffic across multiple paths. Such an equilibrium, however, can be unstable. It can be stabilized by adding a static component to link cost, but at the expense of a reduced utility in equilibrium. If link capacities are optimally provisioned, however, pure static routing, which is necessarily stable, is sufficient to maximize utility. Moreover singlepath routing again achieves the same utility as multipath routing at optimality. Index Terms — Utility optimization, congestion control, TCP
The multiplicative weights update method: a meta algorithm and applications
, 2005
"... Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies ..."
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Cited by 53 (10 self)
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Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1