Related papers: Ellipsoid Fitting Up to a Constant

Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has…

Explicit construction of Ramsey graphs has remained a challenging open problem for a long time. Frankl--Wilson \cite{FW}, Alon \cite{A} and Grolmusz \cite{G2} gave the best explicit constructions of graphs on $m$ vertices with no clique or…

We study the complexity of approximating the permanent of a positive semidefinite matrix $A\in \mathbb{C}^{n\times n}$. 1. We design a new approximation algorithm for $\mathrm{per}(A)$ with approximation ratio $e^{(0.9999 + \gamma)n}$,…

We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern $w$ of length $m$ as a subsequence in a random text of length $n$. The quantity of interest is the…

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear…

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is…

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to…

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector, $(T_{1},…

Let $M_{n}$ denote a random symmetric $n\times n$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values $\pm 1$ with probability $1/2$ each). Resolving a conjecture of Vu, we prove that the…

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one.…

Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$…

We give a deterministic 2^{O(n)} algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This has several interesting consequences including improved deterministic algorithms for volume estimation of convex…

In this paper, we focus on the problem of existence and computing of small and large stable models. We show that for every fixed integer k, there is a linear-time algorithm to decide the problem LSM (large stable models problem): does a…

Maximum likelihood estimation of large Markov-switching vector autoregressions (MS-VARs) can be challenging or infeasible due to parameter proliferation. To accommodate situations where dimensionality may be of comparable order to or…

Let $P(d)$ be the probability that a random 0/1-matrix of size $d \times d$ is singular, and let $E(d)$ be the expected number of 0/1-vectors in the linear subspace spanned by d-1 random independent 0/1-vectors. (So $E(d)$ is the expected…

Frankl--Pach and Erd\H{o}s conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian…

Let $A(x)=A\_0+x\_1A\_1+...+x\_nA\_n$ be a linear matrix, or pencil, generated by given symmetric matrices $A\_0,A\_1,...,A\_n$ of size $m$ with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a…

Many authors have studied the phenomenon of typically Gaussian marginals of high-dimensional random vectors; e.g., for a probability measure on $\R^d$, under mild conditions, most one-dimensional marginals are approximately Gaussian if $d$…

The well-known Koml\'os conjecture states that given $n$ vectors in $\mathbb{R}^d$ with Euclidean norm at most one, there always exists a $\pm 1$ coloring such that the $\ell_{\infty}$ norm of the signed-sum vector is a constant independent…

We investigate the problem of packing identical hard objects on regular lattices in d dimensions. Restricting configuration space to parallel alignment of the objects, we study the densest packing at a given aspect ratio X. For rectangles…