results for au:Sedgwick_E in:cs

- We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. This stands in contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in $\mathbb{R}^3$ like 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.
- Jul 08 2014 cs.DS arXiv:1407.1525v2Let ${\cal T}$ be a triangulation of a set ${\cal P}$ of $n$ points in the plane, and let $e$ be an edge shared by two triangles in ${\cal T}$ such that the quadrilateral $Q$ formed by these two triangles is convex. A \em flip of $e$ is the operation of replacing $e$ by the other diagonal of $Q$ to obtain a new triangulation of ${\cal P}$ from ${\cal T}$. The \em flip distance between two triangulations of ${\cal P}$ is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of ${\cal P}$ is at most $k$, for some given $k \in N$. It is a fundamental and a challenging problem. We present an algorithm for the \sc Flip Distance problem that runs in time $O(n + k \cdot c^{k})$, for a constant $c \leq 2 \cdot 14^{11}$, which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.
- This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together researchers working on various aspects of triangulations and of fostering a closer collaboration within the computational geometry and topology community. Triangulations are highly suitable for computations due to their clear combinatorial structure. As a consequence, they have been successfully employed in discrete algorithms to solve purely theoretical problems in a broad variety of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold topology, geometry, and others). However, due to the large variety of applications, requirements vary from field to field and thus different types of triangulations, different tools, and different frameworks are used in different areas of research. This is why today closely related research areas are sometimes largely disjoint leaving potential reciprocal benefits unused. To address these potentials a workshop on Triangulations was held at Oberwolfach Research Institute in 2012. Since then many new collaborations between researchers of different mathematical communities have been established. Regarding the computational geometry community, the theory of manifolds continues to contribute to advances in more applied areas of the field. Many researchers are interested in fundamental mathematical research about triangulations and thus will benefit from a broad set of knowledge about different research areas using different techniques. We hope that this workshop brought together researchers from many different fields of computational geometry to have fruitful discussions which will lead to new interdisciplinary collaborations and solutions.
- We show that the following algorithmic problem is decidable: given a $2$-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in $\mathbf{R}^3$? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold $X$ into the 3-sphere $S^3$. The main step, which allows us to simplify $X$ and recurse, is in proving that if $X$ can be embedded in $S^3$, then there is also an embedding in which $X$ has a short meridian, i.e., an essential curve in the boundary of $X$ bounding a disk in $S^3\setminus X$ with length bounded by a computable function of the number of tetrahedra of $X$.
- Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.