Related papers: The Planted Orthogonal Vectors Problem

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main…

For an undirected graph G=(V,E), a vertex x \in V separates vertices u and v (where u,v \in V, u \neq v) if their distances to x are not equal. Given an integer parameter k \geq 1, a set of vertices L \subseteq V is a feasible solution if…

In the stochastic online vector balancing problem, vectors $v_1,v_2,\ldots,v_T$ chosen independently from an arbitrary distribution in $\mathbb{R}^n$ arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the norm…

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n, 1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…

Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $\Omega(ndk)$ time when clustering $n$ points in…

In this paper, we develop a distributed algorithm for solving a class of distributed convex optimization problems where the local objective functions can be a general nonsmooth function, and all equalities and inequalities are network-wide…

Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…

We consider the problem of estimating the arithmetic average of a finite collection of real vectors stored in a distributed fashion across several compute nodes subject to a communication budget constraint. Our analysis does not rely on any…

Given a simple polygon $P$ consisting of $n$ vertices, we study the problem of designing space-efficient algorithms for computing (i) the visibility polygon of a point inside $P$, (ii) the weak visibility polygon of a line segment inside…

We consider stochastic variational inequalities with monotone operators defined as the expected value of a random operator. We assume the feasible set is the intersection of a large family of convex sets. We propose a method that combines…

The $k$-$\mathtt{means}$++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the $k$-means clustering problem where the goal is to cluster a dataset $\mathcal{X} \subset \mathbb{R} ^d$ into $k$ clusters. The…

A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_1,\varepsilon)$-clusterable if there are $k_1$ balls of radius $\varepsilon$ that cover $S$. A set of vectors $S \subseteq \mathbb{R}^d$ is $(k_2,\delta)$-far from being clusterable if…

We consider the $k$-Center problem and some generalizations. For $k$-Center a set of $k$ center vertices needs to be found in a graph $G$ with edge lengths, such that the distance from any vertex of $G$ to its nearest center is minimized.…

We prove a quantum query lower bound \Omega(n^{(d+1)/(d+2)}) for the problem of deciding whether an input string of size n contains a k-tuple which belongs to a fixed orthogonal array on k factors of strength d<=k-1 and index 1, provided…

In the Densest k-Subgraph problem, given a graph G and a parameter k, one needs to find a subgraph of G induced on k vertices that contains the largest number of edges. There is a significant gap between the best known upper and lower…

We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the…

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the…

We consider distributed optimization on undirected connected graphs. We propose a novel distributed conditional gradient method with (O(1/\sqrt{k})) convergence. Compared with existing methods, each iteration of our method uses both…

The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower…

Optimal transport (OT) defines a powerful framework to compare probability distributions in a geometrically faithful way. However, the practical impact of OT is still limited because of its computational burden. We propose a new class of…