The solution of van der Waerden's problem for permanents

Егорычев, Г. П.

doi:10.1016/0001-8708(81)90044-x

Cited by 149 publications

(147 citation statements)

References 5 publications

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“…Theorem 23 (Egorychev [10], Falikman [11]). Let r ≤ n be positive integers and let B be an r-regular bipartite graph with vertex classes of size n. Then the number of perfect matchings of B is at least ( r n ) n n!.…”

Section: Splitting Into 2-factorsmentioning

confidence: 99%

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Edge-disjoint Hamilton cycles in random graphs

Knox

Kühn

Osthus

2013

Random Struct. Alg.

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Section: Splitting Into 2-factorsmentioning

confidence: 99%

“…Let T = N H (S) ∩ Q , s = |S| and t = |T |. For each vertex x ∈ V (G), let B x = N H (x) ∩ Q and note that (10) x∈S |B x | ≥ εr G s ≥ 16s log 2 n.…”

Section: Proof (I) By Lemma 36 We Havementioning

confidence: 99%

Edge-disjoint Hamilton cycles in random graphs

Knox

Kühn

Osthus

2013

Random Struct. Alg.

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“…The famous van der Waerden conjecture stated that the permanent of nonnegative doubly stochastic matrices is minimized on J, with per( 1 n J) = n!/n n = (1 + o(1))e −n . This conjecture was proved by Friedland [11] with o(e −n ) error term, and exactly by Falikman [9] and Egorychev [6]. It is easy to see that both (13) and (15) evaluate (1 + o(1))e −n on 1 n J.…”

Section: Permanent Of Doubly Stochastic Matricesmentioning

confidence: 77%

Quest for Negative Dependency Graphs

Lü

Mohr

Székely

2012

Springer Proceedings in Mathematics &Amp; Statistics

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The Lovász Local Lemma is a well-known probabilistic technique commonly used to prove the existence of rare combinatorial objects. We explore the lopsided (or negative dependency graph) version of the lemma, which, while more general, appears infrequently in literature due to the lack of settings in which the additional generality has thus far been needed. We present a general framework (matchings in hypergraphs) from which many such settings arise naturally. We also prove a seemingly new generalization of Cayley's formula, which helps defining negative dependency graphs for extensions of forests into spanning trees. We formulate open problems regarding partitions and doubly stochastic matrices that are likely amenable to the use of the lopsided local lemma.

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“…In this case, the system is considered to provide no anonymity as the attacker has determined all input-output message pairings with full certainty. The minimum value of per(P ) is well known to be n!/n n , when all entries in P are 1/n (see, for example, Egorychev [5]). This corresponds to the system providing full anonymity.…”

Section: A Metric For Probabilistic Attacksmentioning

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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The solution of van der Waerden's problem for permanents

Cited by 149 publications

References 5 publications

Edge-disjoint Hamilton cycles in random graphs

Edge-disjoint Hamilton cycles in random graphs

Quest for Negative Dependency Graphs

An Accurate System-Wide Anonymity Metric for Probabilistic Attacks

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