Related papers: Hyperplanes Avoiding Problem and Integer Points Co…
We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a contineus region ($\impre$ model) or a…
We consider the multilinear pagerank problem studied in [Gleich, Lim and Yu, Multilinear Pagerank, 2015], which is a system of quadratic equations with stochasticity and nonnegativity constraints. We use the theory of quadratic vector…
We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials.…
We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…
We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree $d+1$. The polynomials that we investigate are…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Let $\Gamma$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $\Gamma p$, according to…
We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…
$ $In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In $k$-clustering, opening $k$ facilities induces an…
Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$, where the coordinates of the vector $x$ are restricted to take values in some small…
A set of points $X = X_B \cup X_R \subseteq \mathbb{R}^d$ is linearly separable if the convex hulls of $X_B$ and $X_R$ are disjoint, hence there exists a hyperplane separating $X_B$ from $X_R$. Such a hyperplane provides a method for…
For $p$ prime, let $\mathcal{H}^n$ be the linear span of characteristic functions of hyperplanes in $(\mathbb{Z}/p^k\mathbb{Z})^n$. We establish new upper bounds on the dimension of $\mathcal{H}^n$ over $\mathbb{Z}/p\mathbb{Z}$, or…
We present algorithms for the online minimum hitting set problem in geometric range spaces: given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all…
In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…
The main theme of this dissertation is the study of the lattice points in a rational convex polyhedron and their encoding in terms of Barvinok's short rational functions. The first part of this thesis looks into theoretical applications of…
Let $\epsilon_1, \dotsc, \epsilon_n$ be i.i.d. Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity…
In this paper, we consider the problem of finding a maximum cardinality subset of vectors, given a constraint on the normalized squared length of vectors sum. This problem is closely related to Problem 1 from (Eremeev, Kel'manov, Pyatkin,…
A main property of support vector machines consists in the fact that only a small portion of the training data is significant to determine the maximum margin separating hyperplane in the feature space, the so called support vectors. In a…