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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus


CiteScore 2018: 0.80

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1869-6104
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Generic case complexity of the Graph Isomorphism Problem

Gennady A. Noskov / Alexander N. Rybalov
  • Corresponding author
  • Omsk State Technical University, Omsk, Russia; and Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk, Russia
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Published Online: 2016-04-09 | DOI: https://doi.org/10.1515/gcc-2016-0008

Abstract

The edge test is a partial algorithm for the Graph Isomorphism Problem based on comparison the number of edges. We perform a probabilistic analysis of the efficiency of the edge test. With the binomial distribution B(n,p) on the set of inputs, we estimate the asymptotic failure probability of the edge test depending on the rate of decay of parameter p. In particular, if p ≤ 1/2, np → λ > 0, then the asymptotic failure probability is nonzero, so that the edge test does not solve generically the Graph Isomorphism Problem. On the other hand, if p ≤ 1/2, np → ∞, then the failure set is negligible and the edge test generically solves the Graph Isomorphism Problem in polynomial time.

Keywords: Graph isomorphism; binomial random graph; Poisson distribution; Legendre polynomial; generic algorithm

MSC: 05C38; 15A15; 05A15; 15A18

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About the article

Received: 2014-06-08

Published Online: 2016-04-09

Published in Print: 2016-05-01


Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 14-01-00068

Funding Source: DFG

Award identifier / Grant number: SFB 701 of Bielefeld University

Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-11-00085

Gennady Noskov acknowledges financial support from the DFG through SFB 701 of Bielefeld University, Germany and from the Russian Foundation for Basic Research (RFFI-grant 14-01-00068), Russia. Alexander Rybalov acknowledges the Russian Science Foundation (under grant 14-11-00085) for financial support.


Citation Information: Groups Complexity Cryptology, Volume 8, Issue 1, Pages 9–20, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0008.

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