Abstract. In [A. García, M. Noy, and J. Tejel, Comput. Geom., 16 (2000), pp. 211–221], it is shown that a double chain of [Formula: see text] points in the plane admits at least [Formula: see text] polygonizations, and it is claimed that it admits at most [Formula: see text] polygonizations. In this note, we provide a proof of this last result. The proof is based on counting noncrossing path partitions for points in the plane in convex position, where a noncrossing path partition consists of a set of paths connecting the points such that no two edges cross and isolated points are allowed. We prove that a set of [Formula: see text] points in the plane in convex position admits [Formula: see text] noncrossing path partitions and a double chain of [Formula: see text] points in the plane admits at least [Formula: see text] noncrossing path partitions. If isolated points are not allowed, we also show that for [Formula: see text] points in the plane in convex position, there are [Formula: see text] noncrossing path partitions, and for [Formula: see text] points on a double chain, there are at least [Formula: see text] noncrossing path partitions. In addition, using a particular family of noncrossing path partitions for points in convex position, we provide an alternative proof for the result that a double chain of [Formula: see text] points admits at least [Formula: see text] polygonizations.
Abstract. John E. Littlewood posed the question “Is it possible in 3- space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.” Bozóki, Lee, and Rónyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in [Formula: see text] can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in [Formula: see text]) upper bounds on the number of mutually touching cylinders in [Formula: see text]. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and nonequalities.
Abstract. Write [Formula: see text] for the complete [Formula: see text]-graph on [Formula: see text] vertices. For integers [Formula: see text], let [Formula: see text] be the maximum density of [Formula: see text] in [Formula: see text] vertex [Formula: see text]-free [Formula: see text]-graphs. The main contribution of this paper is the upper bound: [Formula: see text]. The graph case ([Formula: see text]) is the first known generalized Turán problem, fully answered by Erdős. The [Formula: see text] case is the hypergraph Turán problem, for which the best-known general upper bound is by de Caen. The result proved here matches both bounds asymptotically, and any triple [Formula: see text] with [Formula: see text] provides a new upper bound. The proof uses techniques from the theory of flag algebras to derive linear relations between different densities. These relations can be combined using linear algebraic methods. Additionally, a simple flag algebraic certificate will be given for [Formula: see text].
Abstract. We investigate fine-grained algorithmic aspects for classical identification problems in graphs, namely, Locating-Dominating Set, and in set systems, namely, Test Cover. In the first problem, an input is a graph [Formula: see text] on [Formula: see text] vertices and an integer [Formula: see text], and the objective is to decide whether there is a subset [Formula: see text] of [Formula: see text] vertices such that any two distinct vertices not in [Formula: see text] are dominated by distinct subsets of [Formula: see text]. In the second problem, an input is a set [Formula: see text] of items, a collection [Formula: see text] of subsets of [Formula: see text] called tests, and an integer [Formula: see text], and the objective is to select a set [Formula: see text] of at most [Formula: see text] tests such that any two distinct items are contained in a distinct subset of tests of [Formula: see text]. For our first result, we Locating-Dominating Set (respectively, Test Cover) parameterized by the treewidth of the input graph (respectively, the incidence graph) does not admit an algorithm running in time [Formula: see text] (respectively, [Formula: see text]), unless the exponential-time hypothesis ( ETH) fails. Next, we prove that unless the ETH fails, Locating-Dominating Set does not admit an algorithm running in time [Formula: see text] or a polynomial-time kernelization algorithm that reduces the solution size and outputs a kernel with [Formula: see text] vertices, and Test Cover does not admit an algorithm running in time [Formula: see text] or a kernel with [Formula: see text] vertices. Again, we show that these lower bounds are tight by designing (kernelization) algorithms with matching running times.
Abstract. Estimating the linear dimensionality of a data set in the presence of noise is a common problem. However, data may also be corrupted by monotone nonlinear distortion that preserves the ordering of matrix entries but causes linear methods for estimating rank to fail. In light of this, we consider the problem of computing the monotone rank, which is the lowest rank consistent with the ordering within columns. We show that each matrix of monotone rank [Formula: see text] corresponds to a point arrangement and a hyperplane arrangement in [Formula: see text], and that the ordering within columns of the matrix can be used to recover information about these arrangements. Using Radon’s theorem and the related concept of the VC-dimension, we can obtain lower bounds on the monotone rank of a matrix. However, we also show that the monotone rank of a matrix can exceed these bounds. In order to obtain better bounds on monotone rank, we develop the connection between monotone rank estimation and oriented matroid theory. Using this connection, we show that monotone rank is difficult to compute: the problem of deciding whether a matrix has monotone rank two is already NP-hard. However, we introduce an “oriented matroid completion” problem as a combinatorial relaxation of the monotone rank problem and show that checking whether a set of sign vectors has matroid completion rank two is easy.
