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Register a new userNegative Instance for the Edge Patrolling Beacon Problem
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Added by
Jayson Lynch
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Can an infinite-strength magnetic beacon always ``catch an iron ball, when the beacon is a point required to be remain nonstrictly outside a polygon, and the ball is a point always moving instantaneously and maximally toward the beacon subject to staying nonstrictly within the same polygon? Kouhestani and Rappaport [JCDCG 2017] gave an algorithm for determining whether a ball-capturing beacon strategy exists, while conjecturing that such a strategy always exists. We disprove this conjecture by constructing orthogonal and general-position polygons in which the ball and the beacon can never be united.
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We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.
Recently, the duals of Federers curvature measures, called dual curvature measures, were discovered by Huang, Lutwak, Yang, and Zhang (ACTA, 2016). In the same paper, they posed the dual Minkowski problem, the characterization problem for dual curvature measures, and proved existence results when the index, q, is in (0,n). The dual Minkowski problem includes the Aleksandrov problem (q = 0) and the logarithmic Minkowski problem (q = n) as special cases. In the current work, a complete solution to the dual Minkowski problem whenever q < 0, including both existence and uniqueness, is presented.
Given a set of points $P$ and axis-aligned rectangles $mathcal{R}$ in the plane, a point $p in P$ is called emph{exposed} if it lies outside all rectangles in $mathcal{R}$. In the emph{max-exposure problem}, given an integer parameter $k$, we want to delete $k$ rectangles from $mathcal{R}$ so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in $mathcal{R}$ are translates of two fixed rectangles. However, if $mathcal{R}$ only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For range space defined by general rectangles, we present a simple $O(k)$ bicriteria approximation algorithm; that is by deleting $O(k^2)$ rectangles, we can expose at least $Omega(1/k)$ of the optimal number of points.
Given two sets of points in the plane, $P$ of $n$ terminals and $S$ of $m$ Steiner points, a Steiner tree of $P$ is a tree spanning all points of $P$ and some (or none or all) points of $S$. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, $P$ and $S$, and a positive integer $k le m$, find a bottleneck Steiner tree of $P$ with at most $k$ Steiner points. The problem has application in the design of wireless communication networks. We first show that the problem is NP-hard and cannot be approximated within factor $sqrt{2}$, unless $P=NP$. Then, we present a polynomial-time approximation algorithm with performance ratio 2.
Given a set P of n points in the plane, a unit-disk graph G_{r}(P) with respect to a radius r is an undirected graph whose vertex set is P such that an edge connects two points p, q in P if the Euclidean distance between p and q is at most r. The length of any path in G_r(P) is the number of edges of the path. Given a value lambda>0 and two points s and t of P, we consider the following reverse shortest path problem: finding the smallest r such that the shortest path length between s and t in G_r(P) is at most lambda. It was known previously that the problem can be solved in O(n^{4/3} log^3 n) time. In this paper, we present an algorithm of O(lfloor lambda rfloor cdot n log n) time and another algorithm of O(n^{5/4} log^2 n) time.
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