[Paper Review] Graph isomorphisms in quasi-polynomial time

H. A. Helfgott, Jitendra Bajpai+1 more|arXiv (Cornell University)|Oct 12, 2017

Advanced Graph Theory Research8 references21 citations

TL;DR

This paper presents a quasi-polynomial time algorithm for solving the graph isomorphism problem, achieving a runtime of exp(O((log n)^O(1))) by leveraging group-theoretic techniques and canonical colorings. The solution builds on Luks' bounded-degree algorithm and uses a novel framework of canonical labelings and local symmetry reduction to achieve the breakthrough result.

ABSTRACT

Let us be given two graphs $Γ_1$, $Γ_2$ of $n$ vertices. Are they isomorphic? If they are, the set of isomorphisms from $Γ_1$ to $Γ_2$ can be identified with a coset $H\cdotπ$ inside the symmetric group on $n$ elements. How do we find $π$ and a set of generators of $H$? The challenge of giving an always efficient algorithm answering these questions remained open for a long time. Babai has recently shown how to solve these problems -- and others linked to them -- in quasi-polynomial time, i.e. in time $\exp\left(O(\log n)^{O(1)} ight)$. His strategy is based in part on the algorithm by Luks (1980/82), who solved the case of graphs of bounded degree.

Motivation & Objective

• To resolve the long-standing open problem of finding an efficient algorithm for determining graph isomorphism.
• To extend Luks' polynomial-time algorithm for bounded-degree graphs to general graphs.
• To develop a framework for canonical labeling that enables efficient isomorphism testing via group actions.
• To reduce the graph isomorphism problem to the string isomorphism problem using symmetric group actions.
• To achieve a runtime improvement from exp(O(√(n log n))) to exp(O((log n)^O(1)))

Proposed method

• Reduces the graph isomorphism problem to the string isomorphism problem via a group action on edge pairs.
• Employs canonical colorings of vertices and edges to detect non-isomorphism and prune the search space.
• Uses the Weisfeiler-Leman refinement process to iteratively refine colorings based on neighborhood structure.
• Applies a divide-and-conquer strategy using stabilizer chains and local symmetry reduction.
• Implements a recursive decomposition based on the structure of the automorphism group and orbit types.
• Introduces a new notion of 'folding' and 'splitting' of graphs to handle high-degree vertices and maintain quasi-polynomial time

Experimental results

Research questions

• RQ1 Can the graph isomorphism problem be solved in quasi-polynomial time for all graphs?
• RQ2 How can canonical colorings be used to efficiently restrict the search space for isomorphisms?
• RQ3 What structural properties of permutation groups allow for efficient isomorphism testing?
• RQ4 Can the time complexity of the graph isomorphism problem be improved beyond the previous exp(O(√(n log n))) bound?
• RQ5 How can Luks' bounded-degree algorithm be generalized to arbitrary graphs using group-theoretic tools?

Key findings

• The graph isomorphism problem can be solved in quasi-polynomial time, specifically exp(O((log n)^O(1))) time.
• The algorithm reduces the graph isomorphism problem to the string isomorphism problem via a symmetric group action on edge sets.
• Canonical colorings based on neighborhood structure and distance information allow for effective pruning of isomorphism candidates.
• The method achieves a significant improvement over the previous best bound of exp(O(√(n log n))) due to Luks.
• The algorithm uses recursive decomposition and stabilizer chains to maintain efficiency across all graph types.
• The correctness of the algorithm was verified and repaired after an initial error in the time analysis, confirming the quasi-polynomial runtime.