## Fast algorithms for approximate semidefinite programming using the multiplicative weights update method (2005)

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Venue: | IN FOCS |

Citations: | 29 - 6 self |

### BibTeX

@INPROCEEDINGS{Arora05fastalgorithms,

author = {Sanjeev Arora and Elad Hazan and Satyen Kale},

title = {Fast algorithms for approximate semidefinite programming using the multiplicative weights update method},

booktitle = {IN FOCS},

year = {2005},

pages = {339--348},

publisher = {}

}

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### Abstract

Semidefinite programming (SDP) relaxations appear in many recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangian-relaxation based technique (modified from the papers of Plotkin, Shmoys, and Tardos (PST), and Klein and Lu) to derive faster algorithms for approximately solving several families of SDP relaxations. The algorithms are based upon some improvements to the PST ideas — which lead to new results even for their framework — as well as improvements in approximate eigenvalue computations by using random sampling.

### Citations

4987 |
Matrix Analysis
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Citation Context ...ij wijEij, where J is the all 1’s matrix, and w0, w1, wij for 1 ≤ i, j ≤ n are non-negative weights summing to 1. To bound the most negative eigenvalue, λn, of C, we use the Gershgorin circle theorem =-=[18]-=-, which implies that |λn| ≤ maxi{ � j |Cij|}. For the matrix C, the dominant contributors to this maximum are the matrices 1 α A and J. For any i, we have � j 1 α |Aij| ≤ 4n since α ≥ 1 4 maxij Aij. A... |

1180 |
Geometric Algorithms and Combinatorial Optimization
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- 1991
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Citation Context ...”. The first polynomial-time algorithm (strictly speaking, an approximation algorithm that computes the solution up to any desired accuracy ε in time polynomial in log 1 ε ) used the Ellipsoid method =-=[16]-=- but faster interior-point methods were later given by Alizadeh [3], and Nesterov and Nemirovskii [24]. The running time of Alizadeh’s algorithm is Õ(√m(m + n3 )L) where L is an input size parameter. ... |

973 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...80 and CCR 0514993. 1 Here and elsewhere in this paper, the Õ notation is used to suppress polylog( mn ε ) factors. Much attention was focused on SDP as a result of the work of Goemans and Williamson =-=[15]-=-, who used SDP to design new approximation algorithms for several NP-hard problems such as MAXCUT, MAX 2-SAT, and MAX 3SAT. In subsequent years, SDP-based approximation algorithms were designed for co... |

499 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ...imation algorithm that computes the solution up to any desired accuracy ε in time polynomial in log 1 ε ) used the Ellipsoid method [16] but faster interior-point methods were later given by Alizadeh =-=[3]-=-, and Nesterov and Nemirovskii [24]. The running time of Alizadeh’s algorithm is Õ(√m(m + n3 )L) where L is an input size parameter. ∗ Supported by David and Lucile Packard Fellowship, NSF grants CCR ... |

455 | The geometry of graphs and some of its algorithmic applications
- Linial, London, et al.
- 1995
(Show Context)
Citation Context ...the time to solve such SDPs is Õ(n4.5 ). In addition to these well-known approximation algorithms, SDP has also proved useful in a host of other settings. For instance, Linial, London, and Rabinovich =-=[23]-=- observe that given an n-point metric space, finding its minimum-distortion embedding into ℓ2 is a SDP with m = O(n 2 ) constraints, which takes Õ(n4 ) time to solve. Recent approximation algorithms f... |

278 |
On Lipschitz embedding of finite metric spaces in Hilbert spaces
- Bourgain
- 1985
(Show Context)
Citation Context ...s to solving the following mathematical program. For convenience of notation, let dij = D 2 ij . min α dij ≤ ||vi − vj|| 2 ≤ α · dij 1 ≤ i < j ≤ n vi ∈ R n 1 ≤ i ≤ n (EMBEDDING) By Bourgain’s theorem =-=[9]-=- the minimum distortion is O(log n). Thus, the optimum value α ∗ of SDP EMBEDDING is O(log 2 n). We assume that the distances are scaled so that � ij dij = n 2 . We claim that this implies that there ... |

278 | Faster and simpler algorithms for multicommodity flow and other fractional packing problems
- Garg, Koenemann
- 1998
(Show Context)
Citation Context ...ein et al [21] showed how to do this for a specific multicommodity flow LP, and Plotkin, Shmoys, and Tardos [25] generalized the method to the family of packing/covering LPs. Later, Garg and Könemann =-=[14]-=- and Fleischer [13] improved the running times further for flow LPs. A recent survey [6] by the authors of the current paper points out that all such algorithms are a subcase of a more general, widely... |

238 | Expander flows, geometric embeddings and graph partitioning
- Arora, Rao, et al.
- 2004
(Show Context)
Citation Context ... In subsequent years, SDP-based approximation algorithms were designed for coloring k-colorable graphs, MAX DICUT, etc. Then progress halted for a few years, until recent work of Arora, Rao, Vazirani =-=[8]-=- that gave a new O( √ log n)-approximation for SPARSEST CUT. The ideas of this paper have been quickly extended to derive similar approximation algorithms for MIN 2CNF DELETION, MIN UNCUT, DIRECTED SP... |

238 | Fast approximation algorithms for fractional packing and covering problems
- Plotkin, Shmoys, et al.
- 1995
(Show Context)
Citation Context ...of exactly. Typically this uses some version of the classical Lagrangian relaxation idea. Klein et al [21] showed how to do this for a specific multicommodity flow LP, and Plotkin, Shmoys, and Tardos =-=[25]-=- generalized the method to the family of packing/covering LPs. Later, Garg and Könemann [14] and Fleischer [13] improved the running times further for flow LPs. A recent survey [6] by the authors of t... |

123 | Fast computation of low-rank matrix approximations
- Achlioptas, Mcsherry
- 2007
(Show Context)
Citation Context ...ces from [22] to our needs (see Lemma 2). Then we suggest further speeding up the Lanczos algorithm by first sparsifying the matrix via random sampling. Our sparsification is quite similar to that of =-=[1]-=-, though we get bounds that are more suitable to our applications. In comparison to [1], our sampling performs better or worse depending on some parameters of the input matrix. The details are in the ... |

122 |
A new algorithm for minimizing convex functions over convex sets
- Vaidya
- 1989
(Show Context)
Citation Context ...first contribution is to modify the MW technique to handle some of these high-width SDPs. Our technique is a hybrid of the MW technique and an “exterior point” (i.e., Ellipsoid-like method) of Vaidya =-=[26]-=-; this lowers the dependence on the width and is very efficient so long as there only “a few” constraints with high width . (Actually the Vaidya algorithm is overkill in most instances, where the numb... |

99 | Assaf Naor, Approximating the Cut–Norm via Grothendieck’s Inequality
- Alon
- 2004
(Show Context)
Citation Context ...oint metric space, finding its minimum-distortion embedding into ℓ2 is a SDP with m = O(n 2 ) constraints, which takes Õ(n4 ) time to solve. Recent approximation algorithms for cut norm of the matrix =-=[4]-=-, and for certain subcases of correlation clustering [10], use a type of SDPs with m = O(n), and hence require time Õ(n 3.5 ). (An intriguing aspect of this work is that the proof that the integrality... |

96 | Approximating fractional multicommodity flow independent of the number of commodities
- Fleischer
(Show Context)
Citation Context ...ed how to do this for a specific multicommodity flow LP, and Plotkin, Shmoys, and Tardos [25] generalized the method to the family of packing/covering LPs. Later, Garg and Könemann [14] and Fleischer =-=[13]-=- improved the running times further for flow LPs. A recent survey [6] by the authors of the current paper points out that all such algorithms are a subcase of a more general, widely useful, and older ... |

92 | Euclidean distortion and the sparsest cut
- Arora, Lee, et al.
- 2005
(Show Context)
Citation Context ...r have been quickly extended to derive similar approximation algorithms for MIN 2CNF DELETION, MIN UNCUT, DIRECTED SPARSEST CUT, and DIRECTED BALANCED SEPARATOR in [2] and NON-UNIFORM SPARSEST CUT in =-=[11, 7]-=-. These new results rely on the so-called triangle inequality constraints, which impose a constraint for every triple of points. Thus the number of constraints m = O(n 3 ), and the time to solve such ... |

84 | Faster approximation algorithms for unit capacity concurrent flow problems with applications to routing and sparsest cuts
- Klein, Plotkin, et al.
- 1994
(Show Context)
Citation Context ... many cases, the same proof of the approximation ratio); (B) Solving the LP approximately instead of exactly. Typically this uses some version of the classical Lagrangian relaxation idea. Klein et al =-=[21]-=- showed how to do this for a specific multicommodity flow LP, and Plotkin, Shmoys, and Tardos [25] generalized the method to the family of packing/covering LPs. Later, Garg and Könemann [14] and Fleis... |

69 |
Maximizing Quadratic Programs: Extending Grothendieck’s Inequality
- Charikar, Wirth
- 2004
(Show Context)
Citation Context ...ing into ℓ2 is a SDP with m = O(n 2 ) constraints, which takes Õ(n4 ) time to solve. Recent approximation algorithms for cut norm of the matrix [4], and for certain subcases of correlation clustering =-=[10]-=-, use a type of SDPs with m = O(n), and hence require time Õ(n 3.5 ). (An intriguing aspect of this work is that the proof that the integrality gap of the SDP used in [4] is O(1) uses the famous Groth... |

56 | The multiplicative weights update method: a meta algorithm and applications. Working paper
- Arora, Hazan, et al.
- 2005
(Show Context)
Citation Context ...Shmoys, and Tardos [25] generalized the method to the family of packing/covering LPs. Later, Garg and Könemann [14] and Fleischer [13] improved the running times further for flow LPs. A recent survey =-=[6]-=- by the authors of the current paper points out that all such algorithms are a subcase of a more general, widely useful, and older framework they called Multiplicative Weights Update method algorithms... |

48 |
O( √ log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
- Agarwal, Charikar, et al.
- 2005
(Show Context)
Citation Context ...SPARSEST CUT. The ideas of this paper have been quickly extended to derive similar approximation algorithms for MIN 2CNF DELETION, MIN UNCUT, DIRECTED SPARSEST CUT, and DIRECTED BALANCED SEPARATOR in =-=[2]-=- and NON-UNIFORM SPARSEST CUT in [11, 7]. These new results rely on the so-called triangle inequality constraints, which impose a constraint for every triple of points. Thus the number of constraints ... |

48 | Estimating the largest eigenvalue by the power and lanczos algorithms with a random start
- Kuczynski, Wozniakowski
- 1992
(Show Context)
Citation Context ... numerical analysts, but has not been used in theory papers thus far because worst-case analysis for it is hard to find in the literature. We adapt an analysis for positive semidefinite matrices from =-=[22]-=- to our needs (see Lemma 2). Then we suggest further speeding up the Lanczos algorithm by first sparsifying the matrix via random sampling. Our sparsification is quite similar to that of [1], though w... |

46 | Sequential and parallel algorithms for mixed packing and covering
- Young
- 2001
(Show Context)
Citation Context ... (i.e. RHS of constraints violated by a at most a factor of 1 ± ε) mixed packing and covering formulations in time proportional to the width squared. For the special case of linear programming, Young =-=[27]-=- provides an algorithm that is independent of the width completely. Recently Jansen [19] obtained an approximation algorithm for general fractional mixed covering and packing problems that is independ... |

44 | Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
- Chawla, Gupta, et al.
(Show Context)
Citation Context ...r have been quickly extended to derive similar approximation algorithms for MIN 2CNF DELETION, MIN UNCUT, DIRECTED SPARSEST CUT, and DIRECTED BALANCED SEPARATOR in [2] and NON-UNIFORM SPARSEST CUT in =-=[11, 7]-=-. These new results rely on the so-called triangle inequality constraints, which impose a constraint for every triple of points. Thus the number of constraints m = O(n 3 ), and the time to solve such ... |

24 |
A semidefinite programming approach to side-chain positioning with new rounding strategies
- Chazelle, Kingsford, et al.
(Show Context)
Citation Context ...at a biological probability estimation problem (HAPLOFREQ), which estimates the frequencies of haplotypes from a noisy sample, can be solved using SDP with m = O(n 2 ). Chazelle, Kingsford, and Singh =-=[12]-=- use an SDP for side chain positioning (SCP), a problem in genomics. Given the growing popularity of SDP, it would be extremely useful to develop alternative approaches that avoidsthe use of general-p... |

23 | Efficient approximation algorithms semidefinite programs arising from max-cut and coloring
- Klein, Lu
- 1996
(Show Context)
Citation Context ...k from [8] has yet to be extended to problems other than uniform SPARSEST CUT, though this may yet happen. Thus improvements of type (A) have not been forthcoming for the other problems. Klein and Lu =-=[20]-=- initiated study of algorithms of type (B) for SDPs. They adapted the PST/MW framework to approximately solve SDPs that arose in the algorithms of Goemans-Williamson and Karger, Motwani, and Sudan. Th... |

20 |
A logn1/2 approximation to sparsest cut
- Arora, Hazan, et al.
- 2004
(Show Context)
Citation Context ...ver, the duality-based framework of ARV also found one use: it was instrumental in the design of a combinatorial, O( √ log n)-approximation algorithm for (uniform) SPARSEST CUT and ran in Õ(n2 ) time =-=[5]-=-, a significant improvement over the Õ(n4.5 ) running time for the interior point algorithm. Interestingly, this algorithm was also derived in the MW framework. However, the dualitybased framework fro... |

12 | HAPLOFREQ - estimating haplotype frequencies efficiently
- Halperin, Hazan
- 2005
(Show Context)
Citation Context ...time Õ(n 3.5 ). (An intriguing aspect of this work is that the proof that the integrality gap of the SDP used in [4] is O(1) uses the famous Grothendieck inequality from analysis.) Halperin and Hazan =-=[17]-=- showed that a biological probability estimation problem (HAPLOFREQ), which estimates the frequencies of haplotypes from a noisy sample, can be solved using SDP with m = O(n 2 ). Chazelle, Kingsford, ... |

9 | Approximation algorithm for the mixed fractional packing and covering problem
- Jansen
(Show Context)
Citation Context ...ering formulations in time proportional to the width squared. For the special case of linear programming, Young [27] provides an algorithm that is independent of the width completely. Recently Jansen =-=[19]-=- obtained an approximation algorithm for general fractional mixed covering and packing problems that is independent of the width, at the expense of an extra factor of m, the number of constraints, in ... |