Isomorphism for random k-uniform hypergraphs

Debsoumya Chakraborti,Alan Frieze,Simi Haber,Mihir Hasabnis

doi:10.1016/j.ipl.2020.106039

Debsoumya Chakraborti, Alan Frieze + Show 2 more

Open Access

https://doi.org/10.1016/j.ipl.2020.106039

Copy DOI

Abstract

We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random k-uniform hypergraph Hn,p;k, for a wide range of p. We also show that it is solvable w.h.p. for random r-regular, k-uniform hypergraphs Hn,r;k,r=O(1).

Highlights

In this note we study the isomorphism problem for two models of random k-uniform hypergraphs, k ≥ 3
The graph isomorphism problem for random graphs is well understood and in this note we extend some of the ideas to hypergraphs
In a canonical labelling we assign a unique label to each vertex of a graph such that labels are invariant under isomorphism

Summary

Introduction

In this note we study the isomorphism problem for two models of random k-uniform hypergraphs, k ≥ 3. The first paper to study graph isomorphism in this context was that of Babai, Erdos and Selkow [12] They considered the model Gn,p where p is a constant independent of n. The running time of the algorithm is O(n2p) q.s. Our first result concerns the random hypergraph Hn,p;k, the random k-uniform hypergraph on vertex set [n] in which each of the possible edges in [n] k occurs independently with probability p. Suppose that k ≥ 3 and p, 1−p ≫ n−(k−2) log n there exists an O(n2k) time algorithm that finds a canonical labeling for Hn,p;k w.h.p. Bollobas [1] and Kucera [10] proved that random regular graphs have canonical labelings w.h.p. We extend the argument of [1] to regular hypergraphs. There is an O(n8/5) time algorithm that finds a canonical labeling for Hn,r;k w.h.p

Proof of Theorem 2

Proof of Theorem 3