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Register a new userAsymmetric Convex Intersection Testing
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Added by
Wolfgang Mulzer
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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 despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before. We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on $n$ and $m$, whose running time depends reasonably on the dimension $d$.
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For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes $G^{text{hom}}(A)$ and $G^{text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{text{hom}}(A)=G^{text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{text{sim}}(A)=G^{text{sim}}(B)$ if and only if $A$ and $B$ are similar.
Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.
Let $mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_mathcal{D}$ for $mathcal{D}$ is the undirected graph with vertex set $mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{rightarrow}_mathcal{D}$ for $mathcal{D}$ is the directed graph with vertex set $mathcal{D}$ in which there is an edge from a disk $D_1 in mathcal{D}$ to a disk $D_2 in mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $mathcal{D}$ and two non-intersecting disks $s, t in mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_mathcal{D}$ or in $G^{rightarrow}_mathcal{D}$ can be found in $O(n^{3/2}text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_ell$ and $L_r$, and a collection $mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $mathcal{D}$, and let $b$ a point in $S$ below all disks of $mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}text{polylog} n)$ expected time.
A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of $n$ geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in $Omega(log n/loglog n)$ and in $Omega(sqrt{log n})$ for disks or squares of different sizes; it is known for general graphs that the worst case is in $Theta(log^2 n)$. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark 90, Raghavan and Spinrad 03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. 18, Bonamy et al. 18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes $mathcal I$ follow the same road. They show that, for every graph $G$ of a large-enough class $mathcal C$, the complement of an even subdivision of $G$ belongs to the intersection class $mathcal I$. Then they conclude invoking the hardness of Maximum Independent Set on the class $mathcal C$, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. 18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.
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