Faster Canonical Forms for Primitive Coherent Configurations

Sun, Xiaorui; Wilmes, John

doi:10.1145/2746539.2746617

Cited by 16 publications

(13 citation statements)

References 34 publications

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“…The first is about coherent configurations (see [Ba81,SuW,Ba15] for the definition and basic properties). Let X = (V ; R) be a homogeneous coherent configuration with n vertices.…”

Section: Discussionmentioning

confidence: 99%

Graph isomorphism in quasipolynomial time [extended abstract]

Babai

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

491

556

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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. *

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“…The first is about coherent configurations (see [Ba81,SuW,Ba15] for the definition and basic properties). Let X = (V ; R) be a homogeneous coherent configuration with n vertices.…”

Section: Discussionmentioning

confidence: 99%

Graph isomorphism in quasipolynomial time [extended abstract]

Babai

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

491

556

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“…As a result, isomorphism for graphs of genus at most g can be tested in time n O((log g) c ) . The definition of t-CR-bounded graphs builds on the Color Refinement algorithm, a simple combinatorial algorithm that iteratively refines a vertex-coloring in an isomorphism-invariant manner and which forms a fundamental algorithmic tool in the context of the Graph Isomorphism Problem (see, e.g., [5,8,9,25,26,37]). We start by formally defining the outcome of the Color Refinement algorithm in the next subsection.…”

Section: Aggregating Local Certificatesmentioning

confidence: 99%

Hypergraph Isomorphism for Groups with Restricted Composition Factors

Neuen¹

2020

Preprint

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We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V , and asking whether there is a permutation γ ∈ Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n + m) O((log d) c ) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time n O(d) m O(1) .As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K 3,h (h ≥ 3) as a minor in time n O((log h) c ) . In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n O((log g) c ) .

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“…In 1981 Babai [2] defined the distinguishing number of a coherent configuration and used it to establish the aforementioned bound for the minimal base size. This notion was later also used in a recent paper by Sun and Wilmes [42]. In the present paper we use a different concept with the same name.…”

Section: Introductionmentioning

confidence: 97%

A proof of Pyber's base size conjecture

Duyan¹,

Halasi

Maróti

2018

Advances in Mathematics

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Building on earlier papers of several authors, we establish that there exists a universal constant c > 0 such that the minimal base size b(G) of a primitive permutation group G of degree n satisfies log |G|/ log n ≤ b(G) < 45(log |G|/ log n) + c. This finishes the proof of Pyber's base size conjecture. The main part of our paper is to prove this statement for affine permutation groups G = V ⋊ H where H ≤ GL(V ) is an imprimitive linear group. An ingredient of the proof is that for the distinguishing number d(G) (in the sense of Albertson and Collins) of a transitive permutation group G of degree n > 1 we have the estimates n |G| < d(G) ≤ 48 n |G|.

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Faster Canonical Forms for Primitive Coherent Configurations

Cited by 16 publications

References 34 publications

Graph isomorphism in quasipolynomial time [extended abstract]

Graph isomorphism in quasipolynomial time [extended abstract]

Hypergraph Isomorphism for Groups with Restricted Composition Factors

A proof of Pyber's base size conjecture

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