Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs

Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries

We examine the task of locating a target region among those induced by intersections of n halfspaces in R^d. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a ϕ-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is Θ(d), which is in sharp contrast to the worst-case teaching complexity of Θ(n). If instead, we consider the average-case learning complexity, the bounds have a dependency on n as Θ(n) for i.i.d. queries and Θ(dlog(n)) for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for ϕ-separable dichotomies, which generalizes the known O(d) bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965)

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Given a set $\Sigma$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $\rho_1,\rho_2,...,\rho_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(\Sigma)$ of $\Sigma$ is $\Theta(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of spheres in $\Sigma$ with radius $\rho_i$. To prove the lower bound, we construct a set of $\Theta(n_1+n_2)$ spheres in $\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $\rho_i$, $i=1,2$, and $\rho_2\ne\rho_1$, such that their convex hull has combinatorial complexity $\Omega(n_1n_2^{\lfloor\frac{d}{2}\rfloor}+n_2n_1^{\lfloor\frac{d}{2}\rfloor})$. Our construction is then generalized to the case where the spheres have $m\ge{}3$ distinct radii. For the upper bound, we reduce the sphere convex hull problem to the problem of computing the worst-case combinatorial complexity of the convex hull of a set of $m$ $d$-dimensional convex polytopes lying on $m$ parallel hyperplanes in $\mathbb{E}^{d+1}$, where $d\ge{}3$ odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the convex hull of a set $\{\mathcal{P}_1,\mathcal{P}_2,...,\mathcal{P}_m\}$ of $m$ $d$-dimensional convex polytopes lying on $m$ parallel hyperplanes of $\mathbb{E}^{d+1}$ is $O(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of vertices of $\mathcal{P}_i$. We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of the convex hull of parallel polytopes (the new proof gives upper bounds for all face numbers of the convex hull of the parallel polytopes

Average-case Complexity of Teaching Convex Polytopes via Halfspace Queries

We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $\phi$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is $\Theta(d)$, which is in sharp contrast to the worst-case teaching complexity of $\Theta(n)$. If instead, we consider the average-case learning complexity, the bounds have a dependency on $n$ as $\Theta(n)$ for i.i.d. queries and $\Theta(d \log(n))$ for actively chosen queries by the learner. Our proof techniques are based on novel insights from computational geometry, which allow us to count the number of convex polytopes and faces in a Euclidean space depending on the arrangement of halfspaces. Our insights allow us to establish a tight bound on the average-case complexity for $\phi$-separable dichotomies, which generalizes the known $\mathcal{O}(d)$ bound on the average number of "extreme patterns" in the classical computational geometry literature (Cover, 1965)

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems