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Detecting and counting copies of permutation patterns are fundamental algorithmic problems, with applications in the analysis of rankings, nonparametric statistics, and property testing tasks such as independence and quasirandomness…

Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least…

We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial…

Randomized quantum algorithms have been proposed in the context of quantum simulation and quantum linear algebra with the goal of constructing shallower circuits than methods based on block encodings. While the algorithmic complexities of…

We overcome two major bottlenecks in the study of low rank approximation by assuming the low rank factors themselves are sparse. Specifically, (1) for low rank approximation with spectral norm error, we show how to improve the best known…

Monotonicity is a simple yet significant qualitative characteristic. We consider the problem of segmenting a sequence in up to K segments. We want segments to be as monotonic as possible and to alternate signs. We propose a quality metric…

We study the problem of efficiently certifying upper bounds on the independence number of $\ell$-uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within…

This article presents a validation of a recently proposed strongly polynomial-time algorithm for the general linear programming problem. The proposed algorithm is an implicit reduction procedure that combines primal and dual linear…

We study the fair k-set selection problem where we aim to select $k$ sets from a given set system such that the (weighted) occurrence times that each element appears in these $k$ selected sets are balanced, i.e., the maximum (weighted)…

Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$…

We revisit the classic Maximum $k$-Coverage problem: Determine the largest number $t$ of elements that can be covered by choosing $k$ sets from a given family $\mathcal{F} = \{S_1,\dots, S_n\}$ of a size-$u$ universe. A notable special case…

In this paper, we apply a Threshold-Decreasing Algorithm to maximize $k$-submodular functions under a matroid constraint, which reduces the query complexity of the algorithm compared to the greedy algorithm with little loss in approximation…

In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are given two strings $X$ and $Y$ of total length $n$, and we are asked to find a pair of maximal-length factors, one of $X$ and the other of $Y$, such that their…

We consider the problem of computing a $k$-sparse approximation to the Fourier transform of a length $N$ signal. Our main result is a randomized algorithm for computing such an approximation (i.e. achieving the $\ell_2/\ell_2$ sparse…

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…

In the last three decades, the $k$-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a…

This paper develops a new framework, \emph{simultaneous saturation}, designed to quantify the size of sets whose elements are simultaneously large. The framework establishes a correspondence between the magnitude of such sets and a system…

Given a large data matrix $A\in\mathbb{R}^{n\times n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}\sim P_0$, or instead $A$ contains a principal submatrix $A_{{\sf…

We study the problem of interpolating a noisy Fourier-sparse signal in the time duration $[0, T]$ from noisy samples in the same range, where the ground truth signal can be any $k$-Fourier-sparse signal with band-limit $[-F, F]$. Our main…

We give a spectral algorithm for decomposing overcomplete order-4 tensors, so long as their components satisfy an algebraic non-degeneracy condition that holds for nearly all (all but an algebraic set of measure $0$) tensors over…