Random Discrete Morse Theory and a New Library of Triangulations

TL;DR

This paper introduces a probabilistic approach to evaluate the complexity of triangulations using discrete Morse theory and presents a new library of challenging examples for testing topological algorithms.

Contribution

It proposes a novel random heuristic for finding optimal discrete Morse functions and introduces a new library of complex triangulations for algorithm testing.

Findings

The random heuristic effectively finds Morse functions even for large inputs.

The measure correlates with the topology and quality of triangulation.

The new library provides more challenging test cases for computational topology.

Abstract

1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. (2) The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its na\"ivet\'e, this approach turns out to be very successful even in the case of huge inputs. (3) In our view the existing libraries of examples in computational topology are `too easy' for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.