On the complexity of nonnegative-matrix scaling

Kalantari, Bahman; Khachiyan, Leonid

doi:10.1016/0024-3795(94)00188-x

Cited by 43 publications

(45 citation statements)

References 15 publications

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“…Sinkhorn [22] shows that this procedure converges to a unique doubly stochastic matrix of the form RM C where R and C are diagonal matrices if M is a positive matrix. Although there are cases in which Sinkhorn balancing does not converge in finite time, many results show that the number of Sinkhorn iterations needed to scale a matrix so that row and column sums are 1 ± ǫ is polynomial in 1/ǫ [1,17,18].…”

Section: Considering Permutationsmentioning

confidence: 99%

Complexity of combinatorial market makers

Chen

Fortnow

Lambert

et al. 2008

Proceedings of the 9th ACM Conference on Electronic Commerce

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We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson's popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is #P-hard, even when the same language admits polynomial-time matching without the market maker. We then propose an approximation technique for pricing permutation markets based on a recent algorithm for online permutation learning. The connections we draw between LMSR pricing and the vast literature on online learning with expert advice may be of independent interest.

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Section: Considering Permutationsmentioning

confidence: 99%

Complexity of combinatorial market makers

Chen

Fortnow

Lambert

et al. 2008

Proceedings of the 9th ACM Conference on Electronic Commerce

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“…Matrix scaling and, we believe, n-tuple scaling as well, are important problems, even without their ties to permanents and mixed discriminants. Matrix scaling problems were solved via a convex programming approach in [20] and, in a more general setting, in [22].…”

Section: An Overview Of the Mixed Discriminant Approximation Algorithmmentioning

confidence: 99%

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

Gurvits¹,

Samorodnitsky²

2002

Discrete Comput Geom

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Abstract. We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n-tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n-tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in R n to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A "side benefit" is a generalization of Rado's theorem on the existence of a linearly independent transversal.

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“…So in general, this procedure requires an infinite number of iterations. A number of efficient algorithms have therefore been considered, as in Kalantari and Khachiyan [10] and Linial, Samorodnitsky and Wigderson [11], for producing in a finite number of steps, approximate solutions that are within acceptable error bounds.…”

Section: Theorem 1 For Any N × N Biadjacency Matrix a Resulting Frommentioning

confidence: 99%

An Accurate System-Wide Anonymity Metric for Probabilistic Attacks

Bagai

et al. 2011

Privacy Enhancing Technologies

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Abstract.We give a critical analysis of the system-wide anonymity metric of Edman et al. [3], which is based on the permanent value of a doubly-stochastic matrix. By providing an intuitive understanding of the permanent of such a matrix, we show that a metric that looks no further than this composite value is at best a rough indicator of anonymity. We identify situations where its inaccuracy is acute, and reveal a better anonymity indicator. Also, by constructing an information-preserving embedding of a smaller class of attacks into the wider class for which this metric was proposed, we show that this metric fails to possess desirable generalization properties. Finally, we present a new anonymity metric that does not exhibit these shortcomings. Our new metric is accurate as well as general.

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On the complexity of nonnegative-matrix scaling

Cited by 43 publications

References 15 publications

Complexity of combinatorial market makers

Complexity of combinatorial market makers

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

An Accurate System-Wide Anonymity Metric for Probabilistic Attacks

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