The Graph Isomorphism Problem and approximate categories

TL;DR

This paper introduces a novel algebraic approach to the Graph Isomorphism Problem using approximate categories, enabling polynomial-time distinction of complex graph pairs beyond Weisfeiler-Lehman capabilities.

Contribution

It formulates the isomorphism problem as an orbit problem and develops approximate categories that outperform Weisfeiler-Lehman algorithms in distinguishing certain graphs.

Findings

Our algorithms can distinguish Cai-Furer-Immerman graphs in polynomial time.

The new approach generalizes and surpasses Weisfeiler-Lehman algorithms.

The approximate categories provide a new framework for graph isomorphism testing.

Abstract

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C_d(V), d=1,2,3,... whose objects include the elements of V. We show that v_1 and v_2 are not in the same orbit by showing that they are not isomorphic in the category C_d(V) for some d. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can distinguish the Cai-Furer-Immerman graphs in polynomial time.