Graph Similarity Based on Matrix Norms

TL;DR

This paper investigates the computational complexity of graph similarity measures based on matrix norms, revealing strong NP-hardness results even for simple graph classes, and distinguishes between global and local similarity metrics.

Contribution

It formalizes graph similarity measures using matrix norms and proves NP-hardness and inapproximability results for these measures, including for restricted graph classes.

Findings

Global similarity measures relate to Frobenius norm calculations.

Local measures correspond to spectral or operator norms.

NP-hardness results hold even for bounded-degree trees.

Abstract

Quantifying the similarity between two graphs is a fundamental algorithmic problem at the heart of many data analysis tasks for graph-based data. In this paper, we study the computational complexity of a family of similarity measures based on quantifying the mismatch between the two graphs, that is, the "symmetric difference" of the graphs under an optimal alignment of the vertices. An important example is similarity based on graph edit distance. While edit distance calculates the "global" mismatch, that is, the number of edges in the symmetric difference, our main focus is on "local" measures calculating the maximum mismatch per vertex. Mathematically, our similarity measures are best expressed in terms of the adjacency matrices: the mismatch between graphs is expressed as the difference of their adjacency matrices (under an optimal alignment), and we measure it by applying some matrix norm. Roughly speaking, global measures like graph edit distance correspond to entrywise matrix norms like the Frobenius norm and local measures correspond to operator norms like the spectral norm. We prove a number of strong NP-hardness and inapproximability results even for very restricted graph classes such as bounded-degree trees.