Graph Isomorphism in Quasipolynomial Time (1512.03547v2)

Abstract: We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial ($\exp((\log n)^{O(1)})$) time. The best previous bound for GI was $\exp(O(\sqrt{n\log n}))$, where $n$ is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, $\exp(\tilde{O}(\sqrt{n}))$, where $n$ is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks's SI framework and attacks the barrier configurations for Luks's algorithm by group theoretic "local certificates" and combinatorial canonical partitioning techniques. We show that in a well-defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. Luks's barrier situation is characterized by a homomorphism {\phi} that maps a given permutation group $G$ onto $S_k$ or $A_k$, the symmetric or alternating group of degree $k$, where $k$ is not too small. We say that an element $x$ in the permutation domain on which $G$ acts is affected by {\phi} if the {\phi}-image of the stabilizer of $x$ does not contain $A_k$. The affected/unaffected dichotomy underlies the core "local certificates" routine and is the central divide-and-conquer tool of the algorithm.

• The paper introduces a quasipolynomial time algorithm for graph isomorphism, reducing exponential complexity using advanced group-theoretic and combinatorial methods.
• It builds on Luks's framework by employing novel strategies such as the unaffected stabilizer theorem for efficient symmetry breaking.
• The work extends its impact to related problems like string isomorphism and coset intersection, challenging the NP-completeness of graph isomorphism.

An Analysis of "Graph Isomorphism in Quasipolynomial Time"

The paper by László Babai presents a significant advancement in the computational complexity of the Graph Isomorphism (GI) problem by reducing its complexity to quasipolynomial time. This development also extends to related problems such as String Isomorphism (SI) under group action and Coset Intersection (CI). Prior to this research, the most efficient known algorithm for GI required exponential time, specifically $\exp(O(\sqrt{n \log n}))$ as established by Luks in 1983.

Key Contributions

1. Quasipolynomial Time Algorithm: Babai's algorithm solves the GI problem in $\exp((\log n)^{O(1)})$ time. This improvement effectively bridges the gap between the longstanding exponential complexity bound and the elusive polynomial time solution. The algorithm leverages group-theoretic methods and combinatorial techniques to navigate through the computational challenges.
2. Use of Luks's Framework: By building on Luks's classical framework for handling string isomorphisms through permutation group actions, Babai introduces novel strategies to overcome previously insurmountable barrier configurations. This involves utilizing group-theoretic local certificates and combinatorial canonical partitioning.
3. Combinatorial Canonical Partitioning: A major component of the algorithm is the ability to manage obstructions, particularly those posed by the structure of Johnson graphs, which are identified as the sole barriers to effective partitioning in this context.
4. Implications for Related Problems: The reduction to quasipolynomial time is also shown for the SI and CI problems, demonstrating the utility of Babai's techniques beyond just the graph isomorphism domain.
5. Unaffected Stabilizer Theorem: A critical theoretical contribution is the Unaffected Stabilizer Theorem, which provides a divide-and-conquer tool by distinguishing between affected and unaffected elements. This theorem serves as the backbone for the algorithm's recursive structure and is essential for efficient symmetry breaking.

Implications and Future Directions

While the algorithm does not place GI in P, it effectively rules out GI as NP-complete unless quasipolynomial-time NP problems exist. This work represents a substantial step forward in closing the gap between known complexity bounds and practical solvability.

The methods introduced in this paper have potential theoretical impacts, not only in addressing GI but also in exploring the symmetries of complex structures, potentially influencing the development of algorithms for related computational problems in algebraic computation and combinatorics.

Conclusion

This paper marks a significant milestone in understanding the complexity of the Graph Isomorphism problem. Babai's approach, employing advanced group-theoretic concepts alongside combinatorial partitioning methods, opens new avenues for tackling difficult isomorphism problems and may inspire further research towards a complete polynomial-time solution. The broader implications for both theory and computational practice underscore the richness and depth of this research.