Graph isomorphism in quasipolynomial time [extended abstract]

Abstract

We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) and Coset Intersection (CI) can be solved in quasipolynomial (exp((log n) O(1) )) time. The best previous bound for GI was exp(O( √ n log n)), where n is the number of vertices ; 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 address is SI. This problem takes two strings of length n and a permutation group G of degree n (the "ambient group") as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G.Luks's divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks's framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic "local certificates" to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-andconquer. We show that in a well-defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the "local certificates" routine which is based on a new group theoretic result, the "Unaffected stabilizers lemma," that allows us to construct global automorphisms out of local information. *

How Symmetric Are Real-World Graphs? A Large-Scale Study

“…For GI it is even unclear whether it is in N P C (the complexity class of "nondeterministic polynomial-time complete" problems, often informally referred to as the class of the hardest problems to solve) [6]. Either way, efficient algorithms exist [7][8][9][10] and the latest research gathers momentum in lowering the theoretical worst-case complexity [11,12] (still under review as of September 2017).…”

On The Duality Feature of P-Class Problems and NP Complete Problems

“…In 2004, PRIMES is found in P-class and accepted by the research community [10]. And in 2015, Graph isomorphism is reported to have Quasipolynomial time solution [11,12], though the results are still under verification. These show some new perspectives and trends on NPC problems.…”

Assembling a network out of ambiguous patches

“…This is, in general, a computationally expensive procedure even after recent improvements, with the best known algorithm having quasi-polynomial time complexity [3]. With an oracle for the graph isomorphism problem, the average case complexity of this algorithm drops to around |E|(npq 2 ) + |E| 2 , from the subgraph extraction process and the checks for graph isomorphism, respectively.…”