On the complexity of optimal homotopies

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

Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Tightening curves and graphs on surfaces

Any continuous deformation of closed curves on a surface can be decomposed into a finite sequence of local changes on the structure of the curves; we refer to such local operations as homotopy moves. Tightening is the process of deforming given curves into their minimum position; that is, those with minimum number of self-intersections. While such operations and the tightening process has been studied extensively, surprisingly little is known about the quantitative bounds on the number of homotopy moves required to tighten an arbitrary curve. An unexpected connection exists between homotopy moves and a set of local operations on graphs called electrical transformations. Electrical transformations have been used to simplify electrical networks since the 19th century; later they have been used for solving various combinatorial problems on graphs, as well as applications in statistical mechanics, robotics, and quantum mechanics. Steinitz, in his study of 3-dimensional polytopes, looked at the electrical transformations through the lens of medial construction, and implicitly established the connection to homotopy moves; later the same observation has been discovered independently in the context of knots. In this thesis, we study the process of tightening curves on surfaces using homotopy moves and their consequences on electrical transformations from a quantitative perspective. To derive upper and lower bounds we utilize tools like curve invariants, surface theory, combinatorial topology, and hyperbolic geometry. We develop several new tools to construct efficient algorithms on tightening curves and graphs, as well as to present examples where no efficient algorithm exists. We then argue that in order to study electrical transformations, intuitively it is most beneficial to work with monotonic homotopy moves instead, where no new crossings are created throughout the process; ideas and proof techniques that work for monotonic homotopy moves should transfer to those for electrical transformations. We present conjectures and partial evidence supporting the argument

Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary

Computing optimal deformations between two curves is a fundamental question with various applications, and has recently received much attention in both computational topology and in mathematics in the form of homotopies of disks and annular regions. In this paper, we examine this problem in a geometric setting, where we consider the boundary of a polygonal domain with spikes, point obstacles that can be crossed at an additive cost. We aim to continuously morph from one part of the boundary to another, necessarily passing over all spikes, such that the most expensive intermediate curve is minimized, where the cost of a curve is its geometric length plus the cost of any spikes it crosses. We first investigate the general setting where each spike may have a different cost. For the number of inflection points in an intermediate curve, we present a lower bound that is linear in the number of spikes, even if the domain is convex and the two boundaries for which we seek a morph share an endpoint. We describe a 2-approximation algorithm for the general case, and an optimal algorithm for the case that the two boundaries for which we seek a morph share both endpoints, thereby representing the entire boundary of the domain. We then consider the setting where all spikes have the same unit cost and we describe a polynomial-time exact algorithm. The algorithm combines structural properties of homotopies arising from the geometry with methodology for computing Fréchet distances