Edge-disjoint homotopic paths in a planar graph with one hole

AbstractWe prove the following theorem, conjectured by K. Mehlhorn: Let G = (V, E) be a planar graph, embedded in the plane C. Let O denote the interior of the unbounded face, and let I be the interior of some fixed bounded face. Let C1, …, Ck be curves in Cß(I⌣O), with end points in V⌢bd(I⌣O), so that for each vertex v of G the degree of v in G has the same parity as the number of curves Ci beginning or ending in v (counting a curve beginning and ending in v for two). Then there exist pairwise edge-disjoint paths P1, …, Pk in G so that Pi is homotopic to Ci in the space Cß(I⌣O) for i = 1, …, k, if and only if for each dual walk Q from {I, O} to {I, O} the number of edges in Q is not smaller than the number of times Q necessarily intersects the curves Ci. The theorem generalizes a theorem of Okamura and Seymour. We demonstrate how a polynomial-time algorithm finding the paths can be derived

Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

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