# Graph isomorphism in quasipolynomial time [extended abstract]

@article{Babai2016GraphII,
title={Graph isomorphism in quasipolynomial time [extended abstract]},
author={L{\'a}szl{\'o} Babai},
journal={Proceedings of the forty-eighth annual ACM symposium on Theory of Computing},
year={2016}
}
• L. Babai
• Published 19 June 2016
• Mathematics, Computer Science
• Proceedings of the forty-eighth annual ACM symposium on Theory of Computing
We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O~(√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980/82 paper, the problem we actually…
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## References

SHOWING 1-10 OF 84 REFERENCES
Faster Canonical Forms for Strongly Regular Graphs
• Mathematics
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
We show that a canonical form for strongly regular (s. r.) graphs can be found in time exp(Õ(n1/5))and therefore isomorphism of s. r. graphs can be tested within the same time bound, where n is the
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
• Mathematics
SODA
• 2014
It is shown that there are pairs of nonisomorphic n-vertex graphs G and H such that any sum-of-squares (SOS) proof of non isomorphism requires degree Ω(n), and an O(n)-round integrality gap for the Lasserre SDP relaxation is shown.
Monte-Carlo algorithms in graph isomorphism testing
Abstract. We present an O(V 4 log V ) coin flipping algorithm to test vertex-colored graphs with bounded color multiplicities for color-preserving isomorphism. We are also able to generate uniformly
Code equivalence and group isomorphism
• Mathematics
SODA '11
• 2011
A more hopeful program to find a polynomial-time algorithm for semisimple groups, defined as groups without abelian normal subgroups, is initiated and it is proved that the isomorphism problem for this class can be solved in time nO(log log n).
On the Order of Uniprimitive Permutation Groups
One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for the
Faster Canonical Forms for Primitive Coherent Configurations: Extended Abstract
• Mathematics
STOC
• 2015
A new combinatorial structure theory for PCCs is developed that in particular demonstrates the presence of "asymptotically uniform clique geometries" among the constituent graphs of P CCs in a certain range of the parameters.
Graph Isomorphism and the Lasserre Hierarchy
• Computer Science, Mathematics
ArXiv
• 2014
The main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.
Hypergraph isomorphism and structural equivalence of Boolean functions
• E. Luks
• Mathematics, Computer Science
STOC '99
• 1999
It is shown that hypergraph isomorphism can be tested in time O(c”), where n is the sire of the vertex set, and an NC test of equivalence of truth tables under permutation of variables alone is obtained.
Asymptotic Delsarte cliques in distance-regular graphs
• Mathematics
ArXiv
• 2015
We give a new bound on the parameter $$\lambda$$λ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph G, improving and generalizing bounds for strongly regular