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Faster and simpler algorithms for multicommodity flow and other fractional packing problems
"... This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems. ..."
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Cited by 271 (5 self)
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This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems.
Optimal Content Placement for a Largescale VoD System
 In ACM CoNEXT
, 2010
"... IPTV service providers offering VideoonDemand currently use servers at each metropolitan office to store all the videos in their library. With the rapid increase in library sizes, it will soon become infeasible to replicate the entire library at each office. We present an approach for intelligent ..."
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Cited by 20 (2 self)
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IPTV service providers offering VideoonDemand currently use servers at each metropolitan office to store all the videos in their library. With the rapid increase in library sizes, it will soon become infeasible to replicate the entire library at each office. We present an approach for intelligent content placement that scales to large library sizes (e.g., 100Ks of videos). We formulate the problem as a mixed integer program (MIP) that takes into account constraints such as disk space, link bandwidth, and content popularity. To overcome the challenges of scale, we employ a Lagrangian relaxationbased decomposition technique combined with integer rounding. Our technique finds a nearoptimal solution (e.g., within 12%) with orders of magnitude speedup relative to solving even the LP relaxation via standard software. We also present simple strategies to address practical issues such as popularity estimation, content updates, shortterm popularity fluctuation, and frequency of placement updates. Using traces from an operational system, we show that our approach significantly outperforms simpler placement strategies. For instance, our MIPbased solution can serve all requests using only half the link bandwidth used by LRU or LFU cache replacement policies. We also investigate the tradeoff between disk space and network bandwidth. 1.
Approximation algorithms for semidefinite packing problems with applications to MAXCUT and graph coloring
, 2004
"... We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [16] are in this class of problems. We extend a method of Bienstock and Iyengar [4] which was based on ideas from Nesterov [24] to design an al ..."
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Cited by 12 (2 self)
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We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [16] are in this class of problems. We extend a method of Bienstock and Iyengar [4] which was based on ideas from Nesterov [24] to design an algorithm for computing #approximate solutions for this class of semidefinite programs. Our algorithm is in the spirit of Klein and Lu [17], and decreases the dependence of the runtime on # from # 2 to # 1 . For sparse graphs, our method is faster than the best specialized interior point methods. A significant feature of our method is that it treats both the Maxcut and the graph coloring problem in a unified manner. 1
Beating simplex for fractional packing and covering linear programs. FOCS
, 2007
"... We give an approximation algorithm for packing and covering linear programs (linear programs with nonnegative coefficients). Given a constraint matrix with n nonzeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a f ..."
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Cited by 11 (2 self)
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We give an approximation algorithm for packing and covering linear programs (linear programs with nonnegative coefficients). Given a constraint matrix with n nonzeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of OPT (the optimal cost) in time O(n + (r + c)log(n)/ε 2). For dense problems (with r, c = O ( √ n)) the time is O(n + √ n log(n)/ε 2) — linear even as ε → 0. In comparison, previous Lagrangianrelaxation algorithms generally take at least Ω(n log(n)/ε 2) time, while (for small ε) the Simplex algorithm typically takes at least Ω(n min(r, c)) time. 1.
Approximate level method
"... In this paper we propose and analyze a variant of the level method [4], which is an algorithm for minimizing nonsmooth convex functions. The main work per iteration is spent on 1) minimizing a piecewiselinear model of the objective function and on 2) projecting onto the intersection of the feasible ..."
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Cited by 11 (0 self)
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In this paper we propose and analyze a variant of the level method [4], which is an algorithm for minimizing nonsmooth convex functions. The main work per iteration is spent on 1) minimizing a piecewiselinear model of the objective function and on 2) projecting onto the intersection of the feasible region and a polyhedron arising as a level set of the model. We show that by replacing exact computations in both cases by approximate computations, in relative scale, the theoretical iteration complexity increases only by the factor of four. This means that while spending less work on the subproblems, we are able to retain the good theoretical properties of the level method.
Probabilistic Analysis of Linear Programming Decoding
"... Abstract—We initiate the probabilistic analysis of linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high pr ..."
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Cited by 10 (3 self)
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Abstract—We initiate the probabilistic analysis of linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finitelength result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bitflipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting byproduct of our analysis is to establish the existence of “probabilistic expansion ” in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting. Index Terms—Binarysymmetric channel (BSC), channel coding, errorcontrol coding, expanders, factor graphs, linear programming decoding, lowdensity paritycheck (LDPC) codes, randomized algorithms, sum–product algorithm. I.