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We derive the asymptotic distribution of the domination number of a new family of random digraph called proximity catch digraph (PCD), which has application to statistical testing of spatial point patterns and to pattern recognition. The…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are…
We present a dynamic programming algorithm for selecting a representative subset of size $k$ from a given set with $n$ points such that the Riesz $s$-energy is near minimized. While NP-hard in general dimensions, the one-dimensional case…
The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the…
We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a…
The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with…
We study the problem of optimal subset selection from a set of correlated random variables. In particular, we consider the associated combinatorial optimization problem of maximizing the determinant of a symmetric positive definite matrix…
In this paper, we investigate the probabilistic set covering problem (PSCP) in which the right-hand side is a binary random vector and the covering constraint is required to be satisfied with a prespecified probability. We consider the case…
This paper considers structural optimization under a reliability constraint, where the input distribution is only partially known. Specifically, when we only know that the expected value vector and the variance-covariance matrix of the…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with…
Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one…
A result of Spencer states that every collection of $n$ sets over a universe of size $n$ has a coloring of the ground set with $\{-1,+1\}$ of discrepancy $O(\sqrt{n})$. A geometric generalization of this result was given by Gluskin (see…
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…
We study the discrepancy of jittered sampling sets: such a set $\mathcal{P} \subset [0,1]^d$ is generated for fixed $m \in \mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point…
The following class of problems arose out of vain attempts to show that the Pascal's triangle adic transformation has trivial spectrum. Partition a set of size $N$ into sets of size $S \equiv S(N)$ (ignoring leftovers). What is the…
Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the Kernel Density Estimate at a point $x\in\mathbb{R}^{d}$ is defined as $\mathrm{KDE}_{P}(x)=\frac{1}{|P|}\sum_{y\in P} k(x,y)$. We study the problem of designing a data…
We give a polynomial time algorithm to decode multivariate polynomial codes of degree $d$ up to half their minimum distance, when the evaluation points are an arbitrary product set $S^m$, for every $d < |S|$. Previously known algorithms can…
Consider a set $X\subseteq \mathbb{R}^d$ which is 1-dense, namely, it intersects every unit ball. We show that we can get from any point to any other point in $\mathbb{R}^d$ in $n$ steps so that the intermediate points are in $X$, and the…