Related papers: Local Complexity of Polygons
Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a…
We establish how a higher local field can be described as a locally convex vector space once an embedding of a local field into it has been fixed. This extends previous results that had been obtained in the two-dimensional case. In…
We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.
A polynomial depth quantum circuit effects, by definition a poly-local unitary transformation of tensor product state space. It is a physically reasonable belief [Fy][L][FKW] that these are precisely the transformations which will be…
Let $P$ be a set of $n$ points in $\mathbb{R}^2$. For a given positive integer $w<n$, our objective is to find a set $C \subset P$ of points, such that $CH(P\setminus C)$ has the smallest number of vertices and $C$ has at most $n-w$ points.…
In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by…
In this paper, we formulate a simple algorithm that detects contours around a region of interest in an image. After an initial smoothing, the method is based on viewing an image as a topographic surface and finding convex and/or concave…
Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden…
We introduce new geometric and combinatorial criteria that preclude a neural code from being convex, and use them to tackle the classification problem for codes on six neurons. Along the way, we give the first example of a code that is…
We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…
Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) polygon $C$ and a finite set $P$ of points in…
Automating the solutions of multiple network information theory problems, stretching from fundamental concerns such as determining all information inequalities and the limitations of linear codes, to applied ones such as designing coded…
This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…
Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the…
Visibility plays an important role for decision making in cluttered, uncertain environments. This paper considers the problem of identifying optimal hiding spots for an agent against line-of-sight detection by an adversary whose location is…
A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear ``local constraint'' codes to be associated with the edges…
We give polynomial-time dynamic-programming algorithms finding the areas of words in the presentations $\langle a, b \mid a, b \rangle$ and $\langle a, b \mid a^k, b^k; \ k \in \mathbb{N} \rangle$ of the trivial group. In the first of these…
How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$,…
An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have…
In this paper, we want to study the link between the presence of compact objects with some analytic structure and the global geometry of a weakly complete surface. We begin with a brief survey of some now classic results on the local…