A unifying method for the design of algorithms canonizing combinatorial objects

TL;DR

This paper introduces a unified framework for designing canonization algorithms applicable to a wide range of combinatorial objects, leveraging hereditarily finite sets to transfer isomorphism testing techniques to canonization.

Contribution

It presents a general method for canonizing various combinatorial objects, achieving faster algorithms that match the best known isomorphism testing complexities.

Findings

New fastest canonization algorithms for hypergraphs and permutation groups.

Asymptotic running time matches the best known isomorphism algorithms.

Framework applies to diverse combinatorial structures including codes and tree decompositions.

Abstract

We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation groups, tree decompositions, and so on. Our approach allows for a systematic transfer of the techniques that have been developed for isomorphism testing to canonization. We use it to design a canonization algorithm for combinatorial objects in general. This result gives new fastest canonization algorithms with an asymptotic running time matching the best known isomorphism algorithm for the following types of objects: hypergraphs, hypergraphs of bounded color class size, permutation groups (up to permutational isomorphism) and codes that are explicitly given (up to code equivalence).