Regular partitions of hypergraphs and property testing
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Department
Mathematisch-Naturwissenschaftliche Fakultät II
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Abstract
Die Regularitätsmethode für Graphen wurde vor über 30 Jahren von Szemerédi, für den Beweis seines Dichteresultates über Teilmengen der natürlichen Zahlen, welche keine arithmetischen Progressionen enthalten, entwickelt. Grob gesprochen besagt das Regularitätslemma, dass die Knotenmenge eines beliebigen Graphen in konstant viele Klassen so zerlegt werden kann, dass fast alle induzierten bipartiten Graphen quasi-zufällig sind, d.h. sie verhalten sich wie zufällige bipartite Graphen mit derselben Dichte. Das Regularitätslemma hatte viele weitere Anwendungen, vor allem in der extremalen Graphentheorie, aber auch in der theoretischen Informatik und der kombinatorischen Zahlentheorie, und gilt mittlerweile als eines der zentralen Hilfsmittel in der modernen Graphentheorie. Vor wenigen Jahren wurden Regularitätslemmata für andere diskrete Strukturen entwickelt. Insbesondere wurde die Regularitätsmethode für uniforme Hypergraphen und dünne Graphen verallgemeinert. Ziel der vorliegenden Arbeit ist die Weiterentwicklung der Regularitätsmethode und deren Anwendung auf Probleme der theoretischen Informatik. Im Besonderen wird gezeigt, dass vererbbare (entscheidbare) Hypergrapheneigenschaften, das sind Familien von Hypergraphen, welche unter Isomorphie und induzierten Untergraphen abgeschlossen sind, testbar sind. D.h. es existiert ein randomisierter Algorithmus, der in konstanter Laufzeit mit hoher Wahrscheinlichkeit zwischen Hypergraphen, welche solche Eigenschaften haben und solchen die „weit“ davon entfernt sind, unterscheidet.
About 30 years ago Szemerédi developed the regularity method for graphs, which was a key ingredient in the proof of his famous density result concerning the upper density of subsets of the integers which contain no arithmetic progression of fixed length. Roughly speaking, the regularity lemma asserts, that the vertex set of every graph can be partitioned into a constant number of classes such that almost all of the induced bipartite graphs are quasi-random, i.e., they mimic the behavior of random bipartite graphs of the same density. The regularity lemma had have many applications mainly in extremal graph theory, but also in theoretical computer science and additive number theory, and it is considered one of the central tools in modern graph theory. A few years ago the regularity method was extended to other discrete structures. In particular extensions for uniform hypergraphs and sparse graphs were obtained. The main goal of this thesis is the further development of the regularity method and its application to problems in theoretical computer science. In particular, we will show that hereditary, decidable properties of hypergraphs, that are properties closed under isomorphism and vertex removal, are testable. I.e., there exists a randomised algorithm with constant running time, which distinguishes between Hypergraphs displaying the property and those which are “far” from it.
About 30 years ago Szemerédi developed the regularity method for graphs, which was a key ingredient in the proof of his famous density result concerning the upper density of subsets of the integers which contain no arithmetic progression of fixed length. Roughly speaking, the regularity lemma asserts, that the vertex set of every graph can be partitioned into a constant number of classes such that almost all of the induced bipartite graphs are quasi-random, i.e., they mimic the behavior of random bipartite graphs of the same density. The regularity lemma had have many applications mainly in extremal graph theory, but also in theoretical computer science and additive number theory, and it is considered one of the central tools in modern graph theory. A few years ago the regularity method was extended to other discrete structures. In particular extensions for uniform hypergraphs and sparse graphs were obtained. The main goal of this thesis is the further development of the regularity method and its application to problems in theoretical computer science. In particular, we will show that hereditary, decidable properties of hypergraphs, that are properties closed under isomorphism and vertex removal, are testable. I.e., there exists a randomised algorithm with constant running time, which distinguishes between Hypergraphs displaying the property and those which are “far” from it.
Description
Keywords
randomisierte Testalgorithmen, Regularitätslemmata für Graphen, Regularitätslemmata für Hypergraphen, size-Ramsey Zahl, property testing, regularity lemmas for graphs, regularity lemma for hypergraphs, size-Ramsey number
Dewey Decimal Classification
004 Informatik
Citation
Schacht, Mathias.(2010). Regular partitions of hypergraphs and property testing. 10.18452/13975