On the complexity of optimal homotopies

TL;DR

This paper studies the complexity of optimal homotopies on surfaces, proving the problem is in NP, and introduces algorithms and structural results that advance understanding of homotopy-related problems with applications in topology and mesh analysis.

Contribution

It establishes that the Homotopy Height problem is in NP, provides structural theorems for optimal homotopies, and offers approximation algorithms for the problem on surfaces.

Findings

Homotopy Height is in NP.

Optimal homotopies can be characterized structurally.

An O(log n)-approximation algorithm for Homotopy Height on surfaces.

Abstract

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting.