A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

Given a graph $G$ cellularly embedded on a surface $\Sigma$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $\Sigma$ along $G$ yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

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

A Structural Approach to Tree Decompositions of Knots and Spatial Graphs

Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion

On the tree-width of knot diagrams

We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari\~no.Comment: 14 pages, 6 figures. V2: Minor updates to expositio

Shortest path embeddings of graphs on surfaces

The classical theorem of F\'{a}ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of F\'{a}ry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer

The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time $n^{O(k^{1-1/d})}$ or $2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering objectives have a lower bound of $n^{\Omega(k)}$, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than $n^{o(\sqrt{k})}$, and provide a matching upper bound of $n^{O(\sqrt{k})}$. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity

Universal families of arcs and curves on surfaces

The main goal of this paper is to investigate the minimal size of families of curves on surfaces with the following property: a family of simple closed curves $\Gamma$ on a surface realizes all types of pants decompositions if for any pants decomposition of the surface, there exists a homeomorphism sending it to a subset of the curves in $\Gamma$. The study of such universal families of curves is motivated by questions on graph embeddings, joint crossing numbers and finding an elusive center of moduli space. In the case of surfaces without punctures, we provide an exponential upper bound and a superlinear lower bound on the minimal size of a family of curves that realizes all types of pants decompositions. We also provide upper and lower bounds in the case of surfaces with punctures which we can consider labelled or unlabelled, and investigate a similar concept of universality for triangulations of polygons, where we provide bounds which are tight up to logarithmic factors.Comment: v2: Fixed a mistake in one of the lower bound

Degenerate crossing number and signed reversal distance

The degenerate crossing number of a graph is the minimum number of transverse crossings among all its drawings, where edges are represented as simple arcs and multiple edges passing through the same point are counted as a single crossing. Interpreting each crossing as a cross-cap induces an embedding into a non-orientable surface. In 2007, Mohar showed that the degenerate crossing number of a graph is at most its non-orientable genus and he conjectured that these quantities are equal for every graph. He also made the stronger conjecture that this also holds for any loopless pseudotriangulation with a fixed embedding scheme. In this paper, we prove a structure theorem that almost completely classifies the loopless 2-vertex embedding schemes for which the degenerate crossing number equals the non-orientable genus. In particular, we provide a counterexample to Mohar's stronger conjecture, but show that in the vast majority of the 2-vertex cases, the conjecture does hold. The reversal distance between two signed permutations is the minimum number of reversals that transform one permutation to the other one. If we represent the trajectory of each element of a signed permutation under successive reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme with degenerate crossings. Our main result is proved by leveraging this connection and a classical result in genome rearrangement (the Hannenhali-Pevzner algorithm) and can also be understood as an extension of this algorithm when the reversals do not necessarily happen in a monotone order.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

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