arXiv subjects

Arnaud de Mesmay

Publications and source records attributed to Arnaud de Mesmay.

34 records · Page 2Linked to original sources

arXiv2020

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve $γ$, and a collection of disjoint normal curves $Δ$, there is a polynomial-time algorithm to decide if $γ$ lies in the normal subgroup generated by components of $Δ$ in the fundamental group of the surface after attaching the curves to a basepoint.

cs.CG↗

arXiv2020

A Near-Linear Approximation Scheme for Multicuts of Embedded Graphs with a Fixed Number of Terminals

For an undirected edge-weighted graph $G$ and a set $R$ of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from $G$ disconnects each pair in $R$. We provide an algorithm computing a $(1+\varepsilon)$-approximation of the minimum multicut of a graph $G$ in time $(g+t)^{(O(g+t)^3)}\cdot(1/\varepsilon)^{O(g+t)} \cdot n \log n$, where $g$ is the genus of $G$ and $t$ is the number of terminals. This result is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when $g=0$). In order to achieve this, our article leverages on a novel characterization of a minimum multicut as a family of Steiner trees in the universal cover of a surface on which $G$ is embedded. The algorithm heavily relies on topological techniques, and in particular on the use of homotopical tools and computations in covering spaces, which we blend with classic ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs.

cs.DS↗

arXiv2020

Tightening Curves on Surfaces Monotonically with Applications

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms---the cluster and pipe expansions---to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any $k$-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and $ΔY$-transformations for arbitrary integer $k$. Previous algorithms only existed in the planar setting when $k \le 4$, and all of them rely on extensive case-by-case analysis based on different values of $k$. Our algorithm makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.

math.GT↗

arXiv2019

Link Crossing Number is NP-hard

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.

cs.CG↗

arXiv2019

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 Mariño.

math.GT↗

arXiv2018

The unbearable hardness of unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).

math.GT↗

arXiv2018

Embeddability in $\mathbb{R}^3$ is NP-hard

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and co-NP assuming the Generalized Riemann Hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.

math.GT↗

arXiv2017

The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of $n^{Ω(k)}$ 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 $n^{Ω(k)}$, 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.

cs.DS↗

arXiv2017

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 $γ_1$ 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.

cs.CG↗

arXiv2017

Shortest path embeddings of graphs on surfaces

The classical theorem of Fáry 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 Fáry'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.

cs.CG↗

arXiv2016

Finding non-orientable surfaces in 3-manifolds

We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

math.GT↗

arXiv2015

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

Given a graph $G$ cellularly embedded on a surface $Σ$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $Σ$ 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 Rué, Sau and Thilikos, which may be of independent interest.

cs.DS↗

arXiv2015

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g^{3/2}n^{1/2}) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.

math.CO↗

arXiv2014

On the Complexity of Immersed Normal Surfaces

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide a polynomial-time algorithm to test for a local version of this problem.

math.GT↗

arXiv2013

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G_1 and G_2 of the same abstract graph G on an orientable surface S, decide whether G_1 and G_2 are isotopic; in other words, whether there exists a continuous family of embeddings between G_1 and G_2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G_1 (resp., G_2) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G_1 and G_2 are piecewise-linear embeddings in the plane minus a finite set of points; our algorithm runs in O(n^{3/2}\log n) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.

cs.CG↗

arXiv2011

Dimension reduction for finite trees in L_1

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).

math.MG↗