A generalized turán problem and its applications

Lior Gishboliner, Asaf Shapira*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f , there is a graph property whose 1-sided-error query complexity is precisely f (Θ(1/)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem States that for any function f there is a graph property whose 1-sided-error query complexity is f (Θ(1/)) while its 2-sided-error query complexity is only poly(1/). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turán problem: for fixed graphs H andT, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T, H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2ℓ+1). Our third theorem in this paper gives tight bounds for ex(n,Ck,C) for all the remaining values of k and ℓ.

Original languageEnglish
Title of host publicationSTOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
EditorsMonika Henzinger, David Kempe, Ilias Diakonikolas
PublisherAssociation for Computing Machinery
Pages376-389
Number of pages14
ISBN (Electronic)9781450355599
DOIs
StatePublished - 20 Jun 2018
Event50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: 25 Jun 201829 Jun 2018

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/TerritoryUnited States
CityLos Angeles
Period25/06/1829/06/18

Keywords

  • Generalized turán problem
  • Graph property testing

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