Property testing for cyclic groups and beyond

François Le Gall () and Yuichi Yoshida ()
Additional contact information
François Le Gall: The University of Tokyo
Yuichi Yoshida: Kyoto University

Journal of Combinatorial Optimization, 2013, vol. 26, issue 4, No 2, 636-654

Abstract: Abstract This paper studies the problem of testing if an input (Γ,∘), where Γ is a finite set of unknown size and ∘ is a binary operation over Γ given as an oracle, is close to a specified class of groups. Friedl et al. (Proc. of STOC, 2005) have constructed an efficient tester using poly(log|Γ|) queries for the class of abelian groups. We focus in this paper on subclasses of abelian groups, and show that these problems are much harder: Ω(|Γ|1/6) queries are necessary to test if the input is close to a cyclic group, and Ω(|Γ| c ) queries for some constant c are necessary to test more generally if the input is close to an abelian group generated by k elements, for any fixed integer k≥1. We also show that knowledge of the size of the ground set Γ helps only for k=1, in which case we construct an efficient tester using poly(log|Γ|) queries; for any other value k≥2 the query complexity remains Ω(|Γ| c ). All our upper and lower bounds hold for both the edit distance and the Hamming distance. These are, to the best of our knowledge, the first nontrivial lower bounds for such group-theoretic problems in the property testing model and, in particular, they imply the first exponential separations between the classical and quantum query complexities of testing closeness to classes of groups.

Keywords: Property testing; Group-theoretic problems; Quantum computation (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-011-9445-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:26:y:2013:i:4:d:10.1007_s10878-011-9445-8

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-011-9445-8

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().