A Graph-Matching Formulation of the Interleaving Distance between Merge Trees

TL;DR

This paper introduces a new combinatorial formulation of the interleaving distance between merge trees using graph matching, enabling approximation via linear binary optimization and paving the way for future polynomial-time algorithms.

Contribution

It presents a novel graph-matching based formulation of the interleaving distance and develops recursive algorithms for approximation, connecting combinatorial optimization with topological data analysis.

Findings

Implemented algorithms to approximate the interleaving distance

Compared new methods with existing approximation procedures

Found potential for polynomial-time approximation algorithms

Abstract

In this work we study the interleaving distance between merge trees from a combinatorial point of view. We use a particular type of matching between trees to obtain a novel formulation of the distance. With such formulation, we tackle the problem of approximating the interleaving distance by solving linear binary optimization problems in a recursive and dynamical fashion, obtaining lower and upper bounds. We implement those algorithms to compare the outputs with another approximation procedure presented by other authors. We believe that further research in this direction could lead to polynomial time algorithms to approximate the distance and novel theoretical developments on the topic.