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We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. We call this the $Dyck_{k,n}$ problem.…

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise…

We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [12] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global…

The sharpest known high probability generalization bounds for uniformly stable algorithms (Feldman, Vondr\'{a}k, 2018, 2019), (Bousquet, Klochkov, Zhivotovskiy, 2020) contain a generally inevitable sampling error term of order…

Knapsack is one of the most fundamental problems in theoretical computer science. In the $(1 - \epsilon)$-approximation setting, although there is a fine-grained lower bound of $(n + 1 / \epsilon) ^ {2 - o(1)}$ based on the $(\min,…

Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such…

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove…

Given an integer array $A[1..n]$, the Range Minimum Query problem (RMQ) asks to preprocess $A$ into a data structure, supporting RMQ queries: given $a,b\in [1,n]$, return the index $i\in[a,b]$ that minimizes $A[i]$, i.e.,…

A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\mathsf{SAT}$ problem, the…

Recently, in a series of papers [32,38,39,41], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless…

We give a simple deterministic $O(\log K / \log\log K)$ approximation algorithm for the Min-Max Selecting Items problem, where $K$ is the number of scenarios. While our main goal is simplicity, this result also improves over the previous…

Recently, Czumaj et.al. (arXiv 2017) presented a parallel (almost) $2$-approximation algorithm for the maximum matching problem in only $O({(\log\log{n})^2})$ rounds of the massive parallel computation (MPC) framework, when the memory per…

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and…

Submodular functions are set functions mapping every subset of some ground set of size $n$ into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory…

Classically, the edit distance of two length-$n$ strings can be computed in $O(n^2)$ time, whereas an $O(n^{2-\epsilon})$-time procedure would falsify the Orthogonal Vectors Hypothesis. If the edit distance does not exceed $k$, the running…

For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in…

The closest pair problem is a fundamental problem of computational geometry: given a set of $n$ points in a $d$-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this…

We consider a robust formulation, introduced by Krause et al. (2008), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness…

Min-plus product of two $n\times n$ matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a…

The Kannan-Lov\'asz-Simonovits conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and…