Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs

Ryan O'Donnell, John Wright, Chenggang Wu, Yuan Zhou

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Building on work of Cai, Fiirer, and Immerman [18], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n- vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomor- phism requires degree Ω(n). In other words, we show an Ω (n)- round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 - 10-14)- isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are a-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [23] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism is hard. I.e. for every ε > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 - ε) isomorphic from pairs which are not even (1 - ε0)-)- isomorphic for some universal constant ε0)- Along the way we prove a robust asymmetry result for random graphs and hyper- graphs which may be of independent interest.

Original languageEnglish (US)
Title of host publicationProceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
PublisherAssociation for Computing Machinery,
Pages1659-1677
Number of pages19
ISBN (Print)9781611973389
StatePublished - Jan 1 2014
Externally publishedYes
Event25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States
Duration: Jan 5 2014Jan 7 2014

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
CountryUnited States
CityPortland, OR
Period1/5/141/7/14

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Fingerprint Dive into the research topics of 'Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs'. Together they form a unique fingerprint.

Cite this