Related papers: Erd\H{o}s covering systems
Dearing and Zeck presented a dual algorithm for the problem of the minimum covering ball in $\mathbb{R}^n$. Each iteration of their algorithm has a computational complexity of at least $\mathcal O(n^3)$. In this paper we propose a…
We introduce a method for constructing collections of subsets of $\mathbb{R}^{n}$, using an iterated function system, a set $T,$ and a cost function. We refer to these collections as tilings. The special case where $T$ is the central open…
We present the first algorithm for computing class groups and unit groups of arbitrary number fields that provably runs in probabilistic subexponential time, assuming the Extended Riemann Hypothesis (ERH). Previous subexponential algorithms…
Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective…
Covering relations are a topological tool for detecting periodic orbits, symbolic dynamics and chaotic behavior for autonomous ODE. We extend the method of the covering relations onto systems with a time dependent perturbation. As an…
We design a variation of the Karp-Miller algorithm to compute, in a forward manner, a finite representation of the cover (i.e., the downward closure of the reachability set) of a vector addition system with one zero-test. This algorithm…
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the…
The paper addresses the problem of locating sensors with a circular field of view so that a given line segment is under full surveillance, which is termed as the Disc Covering Problem on a Line. The cost of each sensor includes a fixed…
The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the…
The ProbCover method of Yehuda et al. is a well-motivated algorithm for active learning in low-budget regimes, which attempts to "cover" the data distribution with balls of a given radius at selected data points. We demonstrate, however,…
Essential covers were introduced by Linial and Radhakrishnan as a model that captures two complementary properties: (1) all variables must be included and (2) no element is redundant. In their seminal paper, they proved that every essential…
We introduce a $2$-approximation algorithm for the minimum total covering number problem.
We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices…
A set $A$ of vertices in an $r$-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not\in A$ such that, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (1970)…
In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of…
Many learning problems require predicting sets of objects when the number of objects is not known beforehand. Examples include object detection, molecular modeling, and scientific inference tasks such as astrophysical source detection.…
Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…
We study coverage processes in which each draw reveals a subset of $[n]$, and the goal is to determine the expected number of draws until all items are seen at least once. A classical example is the Coupon Collector's Problem, where each…
Covering-based rough set theory is an extension to classical rough set. The main purpose of this paper is to study covering rough sets from a topological point of view. The relationship among upper approximations based on topological spaces…
Theories of rough sets and soft sets are powerful mathematical tools for modelling various types of vagueness. Hybrid model combining a rough set with a soft set which is called soft rough set proposed by Feng et al. [3] in 2010. In this…