# An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

## TL;DR

This paper proves that certain graph isomorphism algorithms based on individualization-refinement require exponential time in the worst case, establishing a fundamental complexity lower bound.

## Contribution

It constructs a family of graphs that necessitate exponential runtime for all variants of individualization-refinement algorithms, including those with heuristics and Weisfeiler-Leman enhancements.

## Key findings

- Constructed graphs require exponential time for individualization-refinement algorithms.
- Lower bounds hold even with heuristics, invariants, and Weisfeiler-Leman enhancements.
- Results apply when automorphism groups are provided to the algorithms.

## Abstract

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.