On the complexity of optimal homotopies

by   Erin Wolf Chambers, et al.

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 γ_1 and γ_2 on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between γ_1 and γ_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 Fréchet 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.


page 1

page 2

page 3

page 4


An Improved Approximation for Packing Big Two-Bar Charts

Recently, we presented a new Two-Bar Charts Packing Problem (2-BCPP), in...

On the complexity of the middle curve problem

For a set of curves, Ahn et al. introduced the notion of a middle curve ...

A 12/7-approximation algorithm for the discrete Bamboo Garden Trimming problem

We study the discrete Bamboo Garden Trimming problem (BGT), where we are...

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

In this paper we prove that the problem of deciding contractibility of a...

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

The bamboo trimming problem considers n bamboo with growth rates h_1, h_...

Tightening Curves on Surfaces Monotonically with Applications

We prove the first polynomial bound on the number of monotonic homotopy ...

Second Best, Third Worst, Fourth in Line

We investigate decomposable combinatorial labeled structures more fully,...

Please sign up or login with your details

Forgot password? Click here to reset