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Szemerédi's Regularity Lemma and Its Applications to Pairwise Clustering and Segmentation
[chapter]

*
Lecture Notes in Computer Science
*

This paper reports a first attempt at applying

doi:10.1007/978-3-540-74198-5_2
fatcat:uog4dpgnkfc7lhjjehmpizfqqi
*Szemerédi's*result*to*computer vision*and*pattern recognition problems. ...*Szemerédi's**regularity**lemma*is a deep result from extremal graph theory which states that every graph can be well-approximated by the union of a constant number of random-like bipartite graphs, called ... We refer*to*[11, 10] for a survey of the (theoretical)*applications*of*Szemerédi's**lemma**and**its*generalizations. The*regularity**lemma*is basically an existence predicate. ...##
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Revealing Structure in Large Graphs: Szemerédi's Regularity Lemma and its Use in Pattern Recognition
[article]

2016
*
arXiv
*
pre-print

In this paper we will provide an overview of the

arXiv:1609.06583v1
fatcat:ci24pl24y5ax5dujr4tjmf75xy
*regularity**lemma**and**its*algorithmic aspects,*and*will discuss*its*relevance in the context of pattern recognition research. ... Roughly,*it*states that every graph can be approximated by the union of a small number of random-like bipartite graphs called*regular*pairs. ... ACKNOWLEDGMENTS We are grateful*to*M. Bolla, H, Reittu*and*F. Bazsó for reading a preliminary version of the paper,*and**to*the anonymous reviewers for their constructive feedback. ...##
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Regular Partitions and Their Use in Structural Pattern Recognition
[article]

2020
*
arXiv
*
pre-print

We first extend an heuristic version of the RL

arXiv:1909.07420v2
fatcat:yconxoyh4ffezb6ahglrdbnruy
*to*improve*its*efficiency*and**its*robustness. We use the proposed algorithm*to*address graph-based*clustering**and*image*segmentation*tasks. ... Finally, we study the linkage among the*regularity**lemma*, the stochastic block model*and*the minimum description length. ... Using the*regularity**lemma*for*pairwise**clustering*Sperotto*and*Pelillo reported arguably the first practical*application*of the*regularity**lemma**and*related algorithms [97] . ...##
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Hamilton decompositions of regular expanders: Applications

2014
*
Journal of combinatorial theory. Series B (Print)
*

In a recent paper, we showed that every sufficiently large

doi:10.1016/j.jctb.2013.10.006
fatcat:6jr47nkcs5fc3gciuqemj6qkne
*regular*digraph G on n vertices whose degree is linear in n*and*which is a robust outexpander has a decomposition into edge-disjoint Hamilton ... dense quasi-random graphs are robust outexpanders; (vi) a verification of the 'very dense' case of a conjecture of Frieze*and*Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii ... Acknowledgements We would like*to*thank John Lapinskas for an idea which led*to*a simplification of the cycle absorbing argument. ...##
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Hamilton decompositions of regular expanders: A proof of Kelly's conjecture for large tournaments

2013
*
Advances in Mathematics
*

We also apply our result

doi:10.1016/j.aim.2013.01.005
fatcat:abtzn4ovljgwxjowcoilviv23y
*to*solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover*and*Punnen as well as Alon, Gutin*and*Krivelevich. ... We show that every sufficiently large*regular*digraph G on n vertices whose degree is linear in n*and*which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. ... Acknowledgements We would like*to*thank John Lapinskas for an idea which led*to*a simplification of the cycle absorbing argument. ...##
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Ramsey Theory Applications

2004
*
Electronic Journal of Combinatorics
*

Relations of Ramsey-type theorems

doi:10.37236/34
fatcat:gxrfo23hzzewjg7rez76d4xx4i
*to*various fields in mathematics are well documented in published books*and*monographs. ... The main objective of this survey is*to*list*applications*mostly in theoretical computer science of the last two decades not contained in these. ... Acknowledgment I am grateful*to*Noga Alon for encouragements*and*fruitful suggestions. I also thank Gyula Károlyi for helpful comments. ...##
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Stable arithmetic regularity in the finite field model

2018
*
Bulletin of the London Mathematical Society
*

This result is an arithmetic analogue of the stable graph

doi:10.1112/blms.12211
fatcat:3yfvfw3vjrb4hlcqwlurbabcj4
*regularity**lemma*proved by Malliaris*and*Shelah. ...*It*is known that in general, the growth of the codimension of H is required*to*be of tower type depending on the degree of uniformity,*and*that one must allow for a small number of non-uniform cosets. ... In the wake of countless successful*applications*of*Szemerédi's**regularity**lemma**to*problems across mathematics*and*theoretical computer science, a first "arithmetic"*regularity**lemma*was proved by Green ...##
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Approximate Hamilton decompositions of robustly expanding regular digraphs
[article]

2013
*
arXiv
*
pre-print

In turn, our result is used as a tool by K\"uhn

arXiv:1206.2810v2
fatcat:kpfpxwfuubc2fcw5zpv5h7m7sy
*and*Osthus*to*prove that any sufficiently large r-*regular*digraph G which has linear degree*and*is a robust outexpander even has a Hamilton decomposition ...*It*also generalises a result of Frieze*and*Krivelevich on approximate Hamilton decompositions of quasirandom (di)graphs. ... Acknowledgements We are extremely grateful*to*Daniela Kühn for helpful discussions throughout the project. ...##
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Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs

2013
*
SIAM Journal on Discrete Mathematics
*

In turn, our result is used as a tool by Kühn

doi:10.1137/120880951
fatcat:sw2x65wog5ce3n5t2iuwb3ziau
*and*Osthus*to*prove that any sufficiently large r-*regular*digraph G which has linear degree*and*is a robust outexpander even has a Hamilton decomposition. ...*It*also generalises a result of Frieze*and*Krivelevich on approximate Hamilton decompositions of quasirandom (di)graphs. ... The diregularity*lemma*. We will use the directed version of*Szemerédi's**regularity**lemma*.*To*state*it*we need some definitions. ...##
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The square of a Hamilton cycle in randomly perturbed graphs
[article]

2022
*
arXiv
*
pre-print

This is known when α > 1/2,

arXiv:2202.05215v2
fatcat:xj3ingvhd5fcfb2i7otgidwep4
*and*we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each α≤ 1/2. ... We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given α∈ (0,1), the union of any n-vertex graph with minimum degree α n*and*the ... However we first need*to*make each*regular*pair super-*regular**and*unbalance some of the*clusters**to*allow an*application*of*Lemma*3.7. ...##
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The bandwidth theorem for locally dense graphs

2020
*
Forum of Mathematics, Sigma
*

G that ensures G contains every r-chromatic graph H on n vertices of bounded degree

doi:10.1017/fms.2020.39
fatcat:orktyn5nfncenludgvqjf4jgwq
*and*of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás*and*Komlós [The Blow-up*Lemma*, Combinatorics, Probability ... ,*and*Computing, 1999]. ... The authors are also grateful*to*the referees for their helpful*and*careful reviews. Conflict of Interest: None. ...##
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A bandwidth theorem for approximate decompositions
[article]

2018
*
arXiv
*
pre-print

Now suppose that G is an n-vertex graph which is close

arXiv:1712.04562v2
fatcat:rfw2bj2fuvfmri262av3va4xna
*to*r-*regular*for some r > (δ_k+o(1))n*and*suppose that H_1,... ... Here a graph is separable if*it*has a sublinear separator whose removal results in a set of components of sublinear size. ... Acknowledgement We are grateful*to*the referee for helpful comments on an earlier version. ...##
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The regularity method for graphs and digraphs
[article]

2014
*
arXiv
*
pre-print

This MSci thesis surveys results in extremal graph theory, in particular relating

arXiv:1406.6531v2
fatcat:blumxwg5azcb7av2jfp4flwqzi
*to*Hamilton cycles.*Szeméredi's**Regularity**Lemma*plays a central role. ... We use Kelly's arguments*to*extend his result*to*any robustly expanding digraph of linear degree. ... Central*to*this project will be*Szeméredi's**Regularity**Lemma*,*Lemma*9. ...##
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A bandwidth theorem for approximate decompositions

2018
*
Proceedings of the London Mathematical Society
*

Acknowledgement We are grateful

doi:10.1112/plms.12218
fatcat:wdolvitxova3pe2lrvrlzh2boe
*to*the referee for helpful comments on an earlier version. ... graphs*and*an n-vertex graph G with e(H) ≤ (1 − o(1))e(G) consisting of super-*regular*pairs,*it*guarantees a packing of H in G (such super-*regular*pairs arise from*applications*of*Szemerédi's**regularity*...*To*resolve this, we apply*Lemma*3.20*to*obtain a suitable 'clique walk' P between Q 1*and*Q i , i.e. the initial sement of P is V (Q 1 ),*its*final*segment*is V (Q i )*and*each*segment*of k consecutive ...##
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Spanning Trees with Few Branch Vertices

2019
*
SIAM Journal on Discrete Mathematics
*

We thank the referees for their careful reading of the paper

doi:10.1137/17m1152759
fatcat:3incjmqvvjagviwqvtlki2rcou
*and*their suggestions for improving the exposition. The first author would like*to*thank Mike Ferrara for introducing him*to*the problem. ...*It*is now standard in nearly spanning subgraph problems*to*use*Szémerédi's**regularity**lemma**to*reduce the problem*to*finding a simpler structure in the so-called reduced graph. ... Now apply*Lemma*6.2*to*G*to*get a t-star cycle C * for some t ≤ s which contains P as a*segment**and*has |A \ V (C * )| = |B \ V (C * )| ≤ ρ 3 n. ...
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