Casting graph isomorphism as a point set registration problem using a simplex embedding and sampling

TL;DR

This paper proposes a novel approach to graph isomorphism by transforming graphs into high-dimensional point sets via simplex embedding, enabling the use of point set registration techniques to determine isomorphism and measure graph similarity.

Contribution

It introduces a new method that reformulates graph isomorphism as a point set registration problem using simplex embedding and sampling, offering a potential new direction for solving the problem.

Findings

Graph isomorphism corresponds to perfect registration of point sets.

Point set registration results can serve as a graph distance measure.

Method extends to automorphism, subgraph isomorphism, and hypergraphs.

Abstract

Graph isomorphism is an important problem as its worst-case time complexity is not yet fully understood. In this study, we try to draw parallels between a related optimization problem called point set registration. A graph can be represented as a point set in enough dimensions using a simplex embedding and sampling. Given two graphs, the isomorphism of them corresponds to the existence of a perfect registration between the point set forms of the graphs. In the case of non-isomorphism, the point set form optimization result can be used as a distance measure between two graphs having the same number of vertices and edges. The related idea of equivalence classes suggests that graph canonization may be an important tool in tackling graph isomorphism problem and an orthogonal transformation invariant feature extraction based on this high dimensional point set representation may be fruitful. The concepts presented can also be extended to automorphism, and subgraph isomorphism problems and can also be applied on hypergraphs with certain modifications.