Regular Flips in mptopcom

TL;DR

This paper explores how to efficiently determine the regularity of triangulations in point configurations by analyzing flip operations, reducing computational effort in enumerating all regular triangulations.

Contribution

It introduces a method to identify flips that preserve regularity, significantly decreasing the number and size of linear programs needed in triangulation enumeration.

Findings

Flip-preserving regularity reduces linear program complexity

Method enables faster enumeration of regular triangulations

Significant computational savings demonstrated

Abstract

A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a triangulation usually is done by solving a linear program. However when checking many flip-connected triangulations for regularity, one can instead ask which flips preserve regularity. When traversing the flip graph for enumerating all regular triangulations, this allows for vast reduction of the linear programs needing to be solved. At the same time the remaining linear programs will be much smaller.