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A simple and sharper proof of the hypergraph Moore bound [article]

Jun-Ting Hsieh, Pravesh K. Kothari, Sidhanth Mohanty
2022 arXiv   pre-print
Their analysis, especially for the case of odd k, is significantly complicated. In this work, we present a substantially simpler and shorter proof of the hypergraph Moore bound.  ...  Our simpler proof also obtains tighter parameters: in particular, the argument gives a new proof of the classical Moore bound of [AHL02] with no loss (the proof in [GKM21] loses a log^3 n factor), and  ...  S.M. would like to thank Siqi Liu and Tselil Schramm for discussions on an independent problem that fueled some of the ideas.  ... 
arXiv:2207.10850v1 fatcat:pn2rux6frrbmdkqqxofpyc32s4

Uniquely $K_r^{(k)}$-Saturated Hypergraphs

András Gyárfás, Stephen G. Hartke, Charles Viss
2018 Electronic Journal of Combinatorics  
This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property.  ...  For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.  ...  A natural question to be asked is that of the sharpness of Theorem 5, or if a sharper upper bound can be obtained.  ... 
doi:10.37236/7534 fatcat:seaio2t7sfexpmra5d3g23icm4

Uniquely K^(k)_r-saturated Hypergraphs [article]

András Gyárfás, Stephen G. Hartke, Charles Viss
2017 arXiv   pre-print
For larger values of n-r the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.  ...  We give a range for n where these hypergraphs exist. For n-r=1 the range is completely determined: k+1< n <(k+2)^2 4.  ...  A natural question to be asked is that of the sharpness of Theorem 5, or if a sharper upper bound can be obtained.  ... 
arXiv:1712.03208v1 fatcat:fiysnvs4qnbdrdlitezvkeoyp4

Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm [article]

Kazusato Oko, Shinsaku Sakaue, Shin-ichi Tanigawa
2022 arXiv   pre-print
This improves the previous bound by Kapralov, Krauthgamer, Tardos, and Yoshida (2021), and it is optimal up to the ε^-1 and log n factors since there is a lower bound of Ω(n^2) even for directed graphs  ...  Our analysis of the undirected case is based on that of Bansal, Svensson, and Trevisan (2019), and the bound matches that of the hypergraph sparsification algorithm by Bansal et al.  ...  Acknowledgements This work was supported by JST ERATO Grant Number JPMJER1903 and JSPS KAKENHI Grant Number 20H05961.  ... 
arXiv:2204.02537v1 fatcat:ui5darnmubevxit4m46fpkomoq

Random k‐SAT: Two Moments Suffice to Cross a Sharp Threshold

Dimitris Achlioptas, Cristopher Moore
2006 SIAM journal on computing (Print)  
In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero.  ...  Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal k-SAT is 2 k−1 ln 2 − O(1).  ...  for discussions on the replica method.  ... 
doi:10.1137/s0097539703434231 fatcat:q2y232grxfaf3mywtvcvt2azte

The history of degenerate (bipartite) extremal graph problems [article]

Zoltán Füredi, Miklós Simonovits
2013 arXiv   pre-print
This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite.  ...  On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.  ...  Acknowledgements The authors are greatly indebted for fruitful discussions and helps to a great number of colleagues, among others to R. Faudree, E. Győri, and Z. Nagy.  ... 
arXiv:1306.5167v2 fatcat:t6puw44f4rayfehkdstmlorsfy

Joins via Geometric Resolutions: Worst-case and Beyond [article]

Mahmoud Abo Khamis, Hung Q. Ngo, Christopher Ré, Atri Rudra
2016 arXiv   pre-print
We present a simple geometric framework for the relational join.  ...  In addition, we use our framework and the same algorithm to show a series of what are colloquially known as beyond worst-case results.  ...  Acknowledgments We thank Paul Beame for clarifying the relation of our notion of geometric resolution with general resolution and we thank Javiel Rojas-Ledesma for bringing Klee's measure problem to our  ... 
arXiv:1404.0703v7 fatcat:g57exkfr7fekffnfwsx77e2xsm

Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limits [article]

Yuetian Luo, Anru R. Zhang
2021 arXiv   pre-print
To our best knowledge, we are the first to give a thorough characterization of the statistical and computational trade-off for such a double computational-barrier problem.  ...  hardness conjectures of hypergraphic planted clique (HPC) detection and hypergraphic planted dense subgraph (HPDS) recovery.  ...  We also thank the Editor, the Associated Editor, and two anonymous referees for their helpful suggestions, which helped improve the presentation and quality of this paper.  ... 
arXiv:2005.10743v3 fatcat:ifi3hgffxrgr7aqogj6falh2fm

Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank [article]

Yi Jiang and Ilya Dumer and Alexey A. Kovalev and Leonid P. Pryadko
2018 arXiv   pre-print
We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds), such that each model can be obtained from the dual of the other after freezing k spins on  ...  limit, and of a high-temperature region where in the ferromagnetic case an extensive homological defect does not affect f(T).  ...  Appendix A: Proof of Theorem 1 Theorem 1.  ... 
arXiv:1805.00644v2 fatcat:antlo62mtjcojlig4bs7fcuuui

Tensor Factor Model Estimation by Iterative Projection [article]

Yuefeng Han, Rong Chen, Dan Yang, Cun-Hui Zhang
2022 arXiv   pre-print
Computational and statistical lower bounds are derived to prove the optimality of the sample size requirement and convergence rate for the proposed methods.  ...  Our algorithms are similar to the higher order orthogonal projection method for tensor decomposition, but with significant differences due to the need to unfold tensors in the iterations and the use of  ...  We would like to thank the Editor, the Associate Editor and the anonymous referees for their detailed reviews, which helped to improve the paper substantially.  ... 
arXiv:2006.02611v2 fatcat:bmf3wkuqtbh7tbuqpqgx3kvn3e

Optimal Bipartite Network Clustering [article]

Zhixin Zhou, Arash A. Amini
2018 arXiv   pre-print
This is further formalized by deriving a minimax lower bound over a class of biclustering problems.  ...  The provable algorithm is derived from a general class of pseudo-likelihood biclustering algorithms that employ simple EM type updates.  ...  A.6 Proofs of Section 3 Proof of Proposition 1. The upper bound has been provided by Corollary 6.  ... 
arXiv:1803.06031v2 fatcat:eiu46m6nuvaozhgjmj4qy7mujq

Colouring random graphs

Colin McDiarmid
1984 Annals of Operations Research  
Improving on a previously-announced bound of Achlioptas and Moore [2] , Kemkes, Pérez-Giménez and Wormald [53] used subgraph conditioning (see [88] ) to prove the following theorem.  ...  Proof As the lower bound follows from the upper bound on α(G n,1/2 ) of Theorem 2.1, it suffices to prove the upper bound.  ...  This implies that the above upper bound on χ t (G) for general G is usually the right asymptotic answer for G n,1/2 . The following is a fuller and more detailed result.  ... 
doi:10.1007/bf01874388 fatcat:efi7gbmiunfyxalbs4wggou73i

14th International Symposium on Mathematical Programming

1990 Mathematical programming  
Finally I report numerical results of a comparison between the variant and a minimization approach using penalty functions.  ...  It is shown that for the success of the variant dom must ful ll a regularity property and that the choice of the normal vectors must meet some demands.Both requirements are ful lled if dom is polyhedral  ...  to a set of linear constraints and simple bounds on the variables.  ... 
doi:10.1007/bf01580875 fatcat:3jtclwmntzgjxkqs5uecombdaa

Community Detection and Stochastic Block Models [article]

Emmanuel Abbe
2022 arXiv   pre-print
The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently.  ...  The monograph gives a principled derivation of the main algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, (linearized  ...  We will break the proof in several parts. A first important result due to [FO05] gives a bound on the norm of AĀ (see [AFWZ17] for a proof): Lemma 9.  ... 
arXiv:1703.10146v2 fatcat:olhrolrol5c67glzmhg42pnyei

Uniform concentration of tensor and neural networks

Alex Christoph Goeßmann, Technische Universität Berlin, Reinhold Schneider, Gitta Kutyniok
2021
In the first part, we develop a unified theoretical framework for the concentration of random variables and the uniform concentration of stochastic processes.  ...  We introduce functionals of stochastic processes and apply them in bounds on the supremum.  ...  Proof. We follow the proof of Theorem 5.12 in [GK20a] and modify in the proof of Theorem 4.4 in [Men16b] the bounds on m ∑ j=1 (κ j ) 2 1 2 and ∥κ∥ L 6 .  ... 
doi:10.14279/depositonce-14763 fatcat:a5ebojsdmrek3mw6v2axwf2mti
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