Hypergraph Matchings and Designs

Keevash, Peter

doi:10.1142/9789813272880_0174

Cited by 9 publications

(7 citation statements)

References 105 publications

(145 reference statements)

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“…In both theory and application, a wide range of significant questions can be recast as existence questions for matchings (see e.g. the books [67,81] and the survey [53]).…”

Section: I41 Cross Matchingsmentioning

confidence: 99%

Global hypercontractivity and its applications

Keevash,

Lifshitz,

Long

et al. 2021

Preprint

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The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the KKL (Kahn-Kalai-Linial) theorem, Friedgut's junta theorem and the invariance principle of Mossel, O'Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform (1/2biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p-biased measures. However, simple examples show that when p is small there is no hypercontractive inequality that is strong enough for such applications.In this paper, we establish an effective hypercontractive inequality for general p that applies to 'global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a p-biased analog of the seminal invariance principle of Mossel, O'Donnell, and Oleszkiewicz.Our sharp threshold results also have significant applications in Extremal Combinatorics. Here we obtain new results on the Turán number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Turán number, answering a question of Mubayi and Verstraëte. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang-Loh-Sudakov conjecture on cross matchings and the Füredi-Jiang-Seiver conjecture on path expansions.

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“…In both theory and application, a wide range of significant questions can be recast as existence questions for matchings (see e.g. the books [67,81] and the survey [53]).…”

Section: I41 Cross Matchingsmentioning

confidence: 99%

Global hypercontractivity and its applications

Keevash,

Lifshitz,

Long

et al. 2021

Preprint

Self Cite

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“…Frankl and Rödl [8] also applied (their version of) this theorem to obtain similar results for other combinatorial problems, for instance the existence of Steiner systems in vector spaces. Keevash [22] raised the meta question of whether there exists a general theorem that provides sufficient conditions for a sparse 'design-like' hypergraph to admit a perfect matching (for a notion of 'design-like' that captures Steiner systems, for example, but hopefully many more structures). Since such hypergraphs will likely be (almost) regular and have small codegree, the existence of an almost perfect matching follows from Pippenger's theorem, and a natural approach would be to use the absorbing method to complete such a matching to a perfect one.…”

Section: Introductionmentioning

confidence: 99%

Pseudorandom hypergraph matchings

Ehard

Glock

Joos

2020

Combinator. Probab. Comp.

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“…Recently there have been works on the randomized analysis of combinatorial design problems [28]- [30]. Motivated by them, we design a randomized heuristic algorithm to solve a Kirkman triple system problem.…”

Section: A Connection To Combinatorial Design Theorymentioning

confidence: 99%

Privacy-Preserving Decentralized Aggregation for Federated Learning

Jeon

Ferdous

Rahman

et al. 2021

IEEE INFOCOM 2021 - IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)

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Federated learning is a promising framework for learning over decentralized data spanning multiple regions. This approach avoids expensive central training data aggregation cost and can improve privacy because distributed sites do not have to reveal privacy-sensitive data. In this paper, we develop a privacy-preserving decentralized aggregation protocol for federated learning. We formulate the distributed aggregation protocol with the Alternating Direction Method of Multiplier (ADMM) and examine its privacy weakness. Unlike prior work that use Differential Privacy or homomorphic encryption for privacy, we develop a protocol that controls communication among participants in each round of aggregation to minimize privacy leakage. We establish its privacy guarantee against an honest-but-curious adversary. We also propose an efficient algorithm to construct such a communication pattern, inspired by combinatorial block design theory. Our secure aggregation protocol based on this novel group communication pattern design leads to an efficient algorithm for federated training with privacy guarantees. We evaluate our federated training algorithm on image classification and next-word prediction applications over benchmark datasets with 9 and 15 distributed sites. Evaluation results show that our algorithm performs comparably to the standard centralized federated learning method while preserving privacy; the degradation in test accuracy is only up to 0.73%.

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Hypergraph Matchings and Designs

Abstract: We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.

Cited by 9 publications

References 105 publications

Global hypercontractivity and its applications

Global hypercontractivity and its applications

Pseudorandom hypergraph matchings

Privacy-Preserving Decentralized Aggregation for Federated Learning

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