On the complexity of optimal homotopies

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 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.1 Introduction This paper considers computational questions pertaining to homotopies: in broad terms, a homotopy between two curves in a topological space is a continuous deformation between these two curves. This can be formalized either in a continuous setting, where it constitutes one of the fundamental constructs of algebraic topology, but also in a more discrete one, where the input is a simplicial, or more generally cellular description of a topological space; this latter setting will be the focus of this article. While considerably more restrictive than the more traditional mathematical settings, this setting

Constructing monotone homotopies and sweepouts

“…However, the exact complexity of this problem remains open, and both papers include a conjecture that the best such morphings will proceed monotonically. The monotonicity result we present in this paper is a key ingredient in showing that this problem lies in the complexity class N P [5].…”

Homotopy Height, Grid-Major Height and Graph-Drawing Height

“…We denote by R(G) the radial graph of G, which is obtained by putting a vertex in the middle of each inner face, connecting each pair of vertices lying on adjacent pairs (vertex,face), removing the inner edges and subdividing each outer edge once. The resulting graph R(G) is a bipartite quadrangulation, and discrete homotopies on R(G) are defined with discrete moves as in G: now the faces of degree 4 can be flipped, and that there are no edge-slides since these do not make sense in a quadrangulation (see e.g.,the introduction of [7] for an inventory of the discrete homotopy moves in general non-triangulated graphs). The outer face is not allowed to be flipped, and for any subdivided outer edge e, we allow a double boundary edge-slide to slide in one step over the two corresponding edges in R(G).…”

“…Conversely, any discrete homotopy of R(G) of height at most 2k between two vertices u and v that are also vertices in G induces a discrete homotopy of G of height at most k between u and v. Indeed, the graph R(G) is bipartite and thus any curve in R(G) can be decomposed into pairs of adjacent edges connecting vertices of G. Each of these pairs can be pushed towards G by doing the reverse of the vibrations described above, and the dictionary between moves of G and R(G) can be read in reverse to extract a discrete homotopy of G of height at most k between u and v. Now the proof of Lemma 4 simply follows from Theorem 4 of [7] showing that optimal homotopies can be assumed to be monotone. More precisely, while Theorem 4 handles homotopies between two cycles forming the boundary of an annulus, the reduction of Proposition 13 in that paper explains how to simply obtain a similar monotonicity property in the setting that we are working with in this paper, where curves are paths with moving endpoints on two boundaries.…”

“…Our results naturally raise the question of the complexity of computing these parameters. Computing the optimal height of homotopies is conjectured but not known to be NP-hard [8]; even arguing that it is in NP is non-trivial [7], although it has a logarithmic approximation [7,21]. Our equalities imply that computing the homotopy-height k = Hh(G) is (non-uniform) fixed-parameter tractable in k. Indeed, it equals grid-major height, which is closed under taking minors.…”

“…We have chosen here one particular variant that seems to be most suitable for graph drawing purposes, and we only study it for triangulated graphs. We refer the reader to other recent works on this parameter [7,8,21] for further discussion.of other variants and their complexity.…”