几何求交图的偶性检验

Asymmetric Convex Intersection Testing

We consider asymmetric convex intersection testing (ACIT). Let $P \subset \mathbb{R}^d$ be a set of $n$ points and $\mathcal{H}$ a set of $n$ halfspaces in $d$ dimensions. We denote by $\text{ch}(P)$ the polytope obtained by taking the convex hull of $P$, and by $\text{fh}(\mathcal{H})$ the polytope obtained by taking the intersection of the halfspaces in $\mathcal{H}$. Our goal is to decide whether the intersection of $\mathcal{H}$ and the convex hull of $P$ are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and …

A Note on Testing Intersection of Convex Sets in Sublinear Time

We present a simple sublinear time algorithm for testing the following geometric property. Let $P_1, ..., P_n$ be $n$ convex sets in $\mathbb{R}^d$ ($n \gg d$), such as polytopes, balls, etc. We assume that the complexity of each set depends only on $d$ (and not on the number of sets $n$). We test the property that there exists a common point in all sets, i.e. that their intersection is nonempty. Our goal is to distinguish between the case where the intersection is nonempty, and the case where even after removing many of the sets the intersection is empty. In particular, our algorithm returns …

线段与$\exists\mathbb{R}$的相交图

Intersection graphs of segments and $\exists\mathbb{R}$

一个顶点集为$\{v_1,v_2,\ldots,v_n\}$的图$G$是一个线段交图，如果在平面中存在线段$s_1,\ldots,s_n$使得$s_i$和$s_j$有公共点当且仅当$\{v_i,v_j\}$是$G$的一条边.在这篇文章中，我们考虑的算法问题，测试是否一个给定的抽象图是一个相交图的部分。事实证明，这个问题是完整的一个有趣的最近推出的一类计算问题，表示为$\exists\mathbb{R}$。这类问题可以在多项式时间内归结为实数上多个变量的多项式不等式系统的可解性。我们讨论了$\exists\mathbb{R}$的定义中的一些微妙之处，我们提供了一个完整的和流线型的帐户证明的$\exists\mathbb{R}$ -完整的识别问题段相交图。在此过程中，我们建立了$\exists\mathbb{R}$ -其他几个问题的完备性。我们还提出了一个决策算法，由于Muchnik，一阶理论的实数。

Clustered planarity testing revisited

The Hanani--Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani--Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati pr…

高效地在RDV图中寻找最大匹配

Finding maximum matchings in RDV graphs efficiently

本文针对RDV图最大匹配问题，提出通过正交射线射击查询和动态线段相交测试的归约方法，实现O(n log n)时间的高效求解。