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In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called $n$-fold integer programming. An $n$-fold integer program (IP) has a highly uniform…
We study the problem of maximizing a monotone submodular function subject to a matroid constraint, and present for it a deterministic non-oblivious local search algorithm that has an approximation guarantee of $1 - 1/e - \varepsilon$ (for…
Change-point problems have appeared in a great many applications for example cancer genetics, econometrics and climate change. Modern multiscale type segmentation methods are considered to be a statistically efficient approach for multiple…
N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer…
We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical…
We study dynamic algorithms for the longest increasing subsequence (\textsf{LIS}) problem. A dynamic \textsf{LIS} algorithm maintains a sequence subject to operations of the following form arriving one by one: (i) insert an element, (ii)…
In this work, we propose an optimization framework for estimating a sparse robust one-dimensional subspace. Our objective is to minimize both the representation error and the penalty, in terms of the l1-norm criterion. Given that the…
We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \,…
We present a randomized algorithm that computes a constant approximation of a graph's arboricity, using $\tilde{O}(n/\lambda)$ queries to adjacency lists and in the same time bound. Here, $n$ and $\lambda$ denote the number of nodes and the…
Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for linear algebra problems. We show that, given a matrix $A \in \R^{n \times d}$ with $n \gg d$ and a $p \in [1, 2)$,…
We give an $\tilde O(n^2)$ time algorithm for computing the exact Dynamic Time Warping distance between two strings whose run-length encoding is of size at most $n$. This matches (up to log factors) the known (conditional) lower bound, and…
In this paper we provide faster algorithms for solving the geometric median problem: given $n$ points in $\mathbb{R}^{d}$ compute a point that minimizes the sum of Euclidean distances to the points. This is one of the oldest non-trivial…
We consider the fundamental problem of maximizing a general quadratic function over an ellipsoidal domain, also known as the trust region problem. We give the first provable linear-time (in the number of non-zero entries of the input)…
We propose a new randomized algorithm for solving convex optimization problems that have a large number of constraints (with high probability). Existing methods like interior-point or Newton-type algorithms are hard to apply to such…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix $N$ becomes very large. There remains…
The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/\epsilon))$ when solving up to $\epsilon$ relative error, is a long-standing open problem in numerical linear algebra…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…
Given a set $Z$ of $n$ positive integers and a target value $t$, the Subset Sum problem asks whether any subset of $Z$ sums to $t$. A textbook pseudopolynomial time algorithm by Bellman from 1957 solves Subset Sum in time $O(nt)$. This has…
The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from…