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

Anna Sperotto, Marcello Pelillo
Lecture Notes in Computer Science  
This paper reports a first attempt at applying 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.  ... 
doi:10.1007/978-3-540-74198-5_2 fatcat:uog4dpgnkfc7lhjjehmpizfqqi

Revealing Structure in Large Graphs: Szemerédi's Regularity Lemma and its Use in Pattern Recognition [article]

Marcello Pelillo, Ismail Elezi, Marco Fiorucci
2016 arXiv   pre-print
In this paper we will provide an overview of the 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.  ... 
arXiv:1609.06583v1 fatcat:ci24pl24y5ax5dujr4tjmf75xy

Regular Partitions and Their Use in Structural Pattern Recognition [article]

Marco Fiorucci
2020 arXiv   pre-print
We first extend an heuristic version of the RL 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] .  ... 
arXiv:1909.07420v2 fatcat:yconxoyh4ffezb6ahglrdbnruy

Hamilton decompositions of regular expanders: Applications

Daniela Kühn, Deryk Osthus
2014 Journal of combinatorial theory. Series B (Print)  
In a recent paper, we showed 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  ...  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.  ... 
doi:10.1016/j.jctb.2013.10.006 fatcat:6jr47nkcs5fc3gciuqemj6qkne

Hamilton decompositions of regular expanders: A proof of Kelly's conjecture for large tournaments

Daniela Kühn, Deryk Osthus
2013 Advances in Mathematics  
We also apply our result 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.  ... 
doi:10.1016/j.aim.2013.01.005 fatcat:abtzn4ovljgwxjowcoilviv23y

Ramsey Theory Applications

Vera Rosta
2004 Electronic Journal of Combinatorics  
Relations of Ramsey-type theorems 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.  ... 
doi:10.37236/34 fatcat:gxrfo23hzzewjg7rez76d4xx4i

Stable arithmetic regularity in the finite field model

C. Terry, J. Wolf
2018 Bulletin of the London Mathematical Society  
This result is an arithmetic analogue of the stable graph 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  ... 
doi:10.1112/blms.12211 fatcat:3yfvfw3vjrb4hlcqwlurbabcj4

Approximate Hamilton decompositions of robustly expanding regular digraphs [article]

Deryk Osthus, Katherine Staden
2013 arXiv   pre-print
In turn, our result is used as a tool by K\"uhn 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.  ... 
arXiv:1206.2810v2 fatcat:kpfpxwfuubc2fcw5zpv5h7m7sy

Approximate Hamilton Decompositions of Robustly Expanding Regular Digraphs

Deryk Osthus, Katherine Staden
2013 SIAM Journal on Discrete Mathematics  
In turn, our result is used as a tool by Kühn 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.  ... 
doi:10.1137/120880951 fatcat:sw2x65wog5ce3n5t2iuwb3ziau

The square of a Hamilton cycle in randomly perturbed graphs [article]

Julia Böttcher, Olaf Parczyk, Amedeo Sgueglia, Jozef Skokan
2022 arXiv   pre-print
This is known when α > 1/2, 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.  ... 
arXiv:2202.05215v2 fatcat:xj3ingvhd5fcfb2i7otgidwep4

The bandwidth theorem for locally dense graphs

Katherine Staden, Andrew Treglown
2020 Forum of Mathematics, Sigma  
G that ensures G contains every r-chromatic graph H on n vertices of bounded degree 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.  ... 
doi:10.1017/fms.2020.39 fatcat:orktyn5nfncenludgvqjf4jgwq

A bandwidth theorem for approximate decompositions [article]

Padraig Condon, Jaehoon Kim, Daniela Kühn, Deryk Osthus
2018 arXiv   pre-print
Now suppose that G is an n-vertex graph which is close 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.  ... 
arXiv:1712.04562v2 fatcat:rfw2bj2fuvfmri262av3va4xna

The regularity method for graphs and digraphs [article]

Amelia Taylor
2014 arXiv   pre-print
This MSci thesis surveys results in extremal graph theory, in particular relating 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.  ... 
arXiv:1406.6531v2 fatcat:blumxwg5azcb7av2jfp4flwqzi

A bandwidth theorem for approximate decompositions

Padraig Condon, Jaehoon Kim, Daniela Kühn, Deryk Osthus
2018 Proceedings of the London Mathematical Society  
Acknowledgement We are grateful 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  ... 
doi:10.1112/plms.12218 fatcat:wdolvitxova3pe2lrvrlzh2boe

Spanning Trees with Few Branch Vertices

Louis DeBiasio, Allan Lo
2019 SIAM Journal on Discrete Mathematics  
We thank the referees for their careful reading of the paper 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.  ... 
doi:10.1137/17m1152759 fatcat:3incjmqvvjagviwqvtlki2rcou
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