Approximating the Cut-Norm via Grothendieck's Inequality

Alon, Noga; Naor, Assaf

doi:10.1137/s0097539704441629

Cited by 209 publications

(383 citation statements)

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“…For general unweighted graphs √ 2|E| = A F < n, where |E| the cardinality of the edge set. Thus, in general, the additive error bound (1) becomes n √ 2|E|, which is an improvement over the previous results of n 2 [1,21]. In addition, from this bound we obtain a PTAS for graphs with |E| = (n 2 ).…”

Section: Summary Of Main Resultsmentioning

confidence: 54%

Sampling Sub-problems of Heterogeneous Max-cut Problems and Approximation Algorithms

2005

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ABSTRACT:Recent work in the analysis of randomized approximation algorithms for NP-hard optimization problems has involved approximating the solution to a problem by the solution of a related subproblem of constant size, where the subproblem is constructed by sampling elements of the original problem uniformly at random. In light of interest in problems with a heterogeneous structure, for which uniform sampling might be expected to yield suboptimal results, we investigate the use of nonuniform sampling probabilities. We develop and analyze an algorithm which uses a novel sampling method to obtain improved bounds for approximating the Max-Cut of a graph. In particular, we show that by judicious choice of sampling probabilities one can obtain error bounds that are superior to the ones obtained by uniform sampling, both for unweighted and weighted versions of Max-Cut. Of at least as much interest as the results we derive are the techniques we use. The first technique is a method to compute a compressed approximate decomposition of a matrix as the product 308DRINEAS, KANNAN, AND MAHONEY of three smaller matrices, each of which has several appealing properties. The second technique is a method to approximate the feasibility or infeasibility of a large linear program by checking the feasibility or infeasibility of a nonuniformly randomly chosen subprogram of the original linear program. We expect that these and related techniques will prove fruitful for the future development of randomized approximation algorithms for problems whose input instances contain heterogeneities.

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Section: Summary Of Main Resultsmentioning

confidence: 54%

Sampling Sub-problems of Heterogeneous Max-cut Problems and Approximation Algorithms

2005

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“…Motivated mainly due to its generic interest and importance, primarily in optimization, the current paper is devoted to the establishment of inequalities of type (1), under various assumptions. Of course such probability estimation cannot hold in general, unless some structures are in place.…”

Section: Introductionmentioning

confidence: 99%

“…the probability bound (1), is sufficiently rich to include some highly nontrivial results beyond optimization as well. As an example, let f (x) = a T x be a linear function, and S = S 0 = B n := {1, −1} n be a binary hypercube.…”

Section: Introductionmentioning

confidence: 99%

“…For every δ ∈ (0, 1 2 ), there is a constant c(δ) > 0 with the following property: Fix a = (a 1 , a 2 , · · · , a n ) T ∈ R n and let ξ 1 , ξ 2 , ..., ξ n be i.i.d. symmetric Bernoulli random variables (taking ±1 with equal probability), then…”

Section: Introductionmentioning

confidence: 99%

“…Besides for any fixed ξ, the problem max y,z∈B n F (ξ, y, z) is a binary constrained bilinear form maximization problem, which admits a deterministic approximation algorithm with approximation ratio 0.03 (see Alon and Naor [1]). Thus we are able to find two vectors y ξ , z ξ ∈ B n in polynomial-time such that F (ξ, y ξ , z ξ ) ≥ 0.03 max y,z∈B n F (ξ, y, z) ≥ 0.03 F (ξ, y * , z * ), which by (3) implies Prob F (ξ, y ξ , z ξ ) ≥ 0.03 δ ln n n F (x * , y * , z * ) ≥ c(δ) n δ .…”

Section: Introductionmentioning

confidence: 99%

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Probability Bounds for Polynomial Functions in Random Variables

Jiang

et al. 2014

Mathematics of OR

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Random sampling is a simple but powerful method in statistics and the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ R n . To do so, one may select a simpler (even finite) subset S 0 ⊆ S, randomly take some samples over S 0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S 0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξ i (1 ≤ i ≤ n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob f (ξ 1 , ξ 2 , · · · , ξ n ) ≥ c 1 n − d 2 F 1 ≥ c 2 , where c 1 , c 2 > 0 are two universal constants. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

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