Robustly self-ordered graphs: Constructions and applications to property testing

Oded Goldreich*, Avi Wigderson*

*Corresponding author for this work

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

3 Scopus citations

Abstract

A graph G is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G = (V, E) is robustly self-ordered if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation π : V → V is proportional to the number of non-fixed-points of π. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust self-ordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sub-logarithmic) connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.

Original languageEnglish
Title of host publication36th Computational Complexity Conference, CCC 2021
EditorsValentine Kabanets
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771931
DOIs
StatePublished - 1 Jul 2021
Externally publishedYes
Event36th Computational Complexity Conference, CCC 2021 - Virtual, Toronto, Canada
Duration: 20 Jul 202123 Jul 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume200
ISSN (Print)1868-8969

Conference

Conference36th Computational Complexity Conference, CCC 2021
Country/TerritoryCanada
CityVirtual, Toronto
Period20/07/2123/07/21

Bibliographical note

Publisher Copyright:
© Oded Goldreich and Avi Wigderson; licensed under Creative Commons License CC-BY 4.0

Keywords

  • Asymmetric graphs
  • Coding theory
  • Expanders
  • Non-malleable extractors
  • Random graphs
  • Testing graph properties
  • Tolerant testing
  • Two-source extractors

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