# Exact computation of the matching distance on 2-parameter persistence modules

## TL;DR

This paper presents a polynomial-time algorithm for exactly computing the matching distance between 2-parameter persistence modules, enhancing the tool's practicality for topological data analysis.

## Contribution

It introduces a novel method to compute the matching distance exactly in polynomial time for 2-parameter modules, using a subdivision of the line space into regions.

## Key findings

- Matching distance can be computed exactly in polynomial time for 2-parameter modules.
- The matching distance is a rational number when input modules have rational bigrades.
- The approach involves subdividing the space of affine lines into regions for analysis.

## Abstract

The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.