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- research-articleSeptember 2024JUST ACCEPTED
Integer programs with bounded subdeterminants and two nonzeros per row
We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the ...
- research-articleSeptember 2024JUST ACCEPTED
- research-articleJuly 2024JUST ACCEPTED
- research-articleMarch 2022
Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)
Journal of the ACM (JACM), Volume 69, Issue 2Article No.: 12, Pages 1–40https://doi.org/10.1145/3505584 - research-articleDecember 2021
Smooth Approximation of Lipschitz Maps and Their Subgradients
Journal of the ACM (JACM), Volume 69, Issue 1Article No.: 8, Pages 1–32https://doi.org/10.1145/3481805We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new ...
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- research-articleApril 2020
Differential Equation Invariance Axiomatization
Journal of the ACM (JACM), Volume 67, Issue 1Article No.: 6, Pages 1–66https://doi.org/10.1145/3380825This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic ...
- research-articleJanuary 2020
Matrix Multiplication, a Little Faster
Journal of the ACM (JACM), Volume 67, Issue 1Article No.: 1, Pages 1–31https://doi.org/10.1145/3364504Strassen’s algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved the leading coefficient of its complexity from 6 to 7. There have been many subsequent asymptotic improvements. Unfortunately, most of these ...
- research-articleJuly 2019
Index Reduction for Differential-algebraic Equations with Mixed Matrices
Journal of the ACM (JACM), Volume 66, Issue 5Article No.: 35, Pages 1–34https://doi.org/10.1145/3341499Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to ...
- research-articleJune 2019
Nonhomogeneous Place-dependent Markov Chains, Unsynchronised AIMD, and Optimisation
Journal of the ACM (JACM), Volume 66, Issue 4Article No.: 24, Pages 1–37https://doi.org/10.1145/3312741A stochastic algorithm is presented for a class of optimisation problems that arise when a group of agents compete to share a single constrained resource in an optimal manner. The approach uses intermittent single-bit feedback, which indicates a ...
- research-articleNovember 2018
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing: The Ideal Proof System
Journal of the ACM (JACM), Volume 65, Issue 6Article No.: 37, Pages 1–59https://doi.org/10.1145/3230742We introduce a new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatzë, proof complexity, ...
- research-articleOctober 2017
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
Journal of the ACM (JACM), Volume 64, Issue 6Article No.: 38, Pages 1–76https://doi.org/10.1145/3127496The outcomes of this article are twofold.
Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in ...
- research-articleMay 2017
Analysis of a Classical Matrix Preconditioning Algorithm
Journal of the ACM (JACM), Volume 64, Issue 2Article No.: 9, Pages 1–23https://doi.org/10.1145/2988227We study a classical iterative algorithm for balancing matrices in the L∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett 8 Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical ...
- research-articleJanuary 2017
Low-Rank Approximation and Regression in Input Sparsity Time
Journal of the ACM (JACM), Volume 63, Issue 6Article No.: 54, Pages 1–45https://doi.org/10.1145/3019134We design a new distribution over m × n matrices S so that, for any fixed n × d matrix A of rank r, with probability at least 9/10, ∥SAx∥2 = (1 ± ε)∥Ax∥2 simultaneously for all x ∈ Rd. Here, m is bounded by a polynomial in rε− 1, and the parameter ε ∈ (...
- research-articleJanuary 2017
Faster Polynomial Multiplication over Finite Fields
Journal of the ACM (JACM), Volume 63, Issue 6Article No.: 52, Pages 1–23https://doi.org/10.1145/3005344Polynomials over finite fields play a central role in algorithms for cryptography, error correcting codes, and computer algebra. The complexity of multiplying such polynomials is still a major open problem. Let p be a prime, and let Mp(n) denote the bit ...
- research-articleMay 2016
Almost Optimal Local Graph Clustering Using Evolving Sets
Journal of the ACM (JACM), Volume 63, Issue 2Article No.: 15, Pages 1–31https://doi.org/10.1145/2856030Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A local graph partitioning algorithm finds a ...
- research-articleMay 2016
A Combinatorial, Primal-Dual Approach to Semidefinite Programs
Journal of the ACM (JACM), Volume 63, Issue 2Article No.: 12, Pages 1–35https://doi.org/10.1145/2837020Semidefinite programs (SDPs) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number ...
- research-articleDecember 2015
Solving Linear Programs without Breaking Abstractions
Journal of the ACM (JACM), Volume 62, Issue 6Article No.: 48, Pages 1–26https://doi.org/10.1145/2822890We show that the ellipsoid method for solving linear programs can be implemented in a way that respects the symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make ...
- research-articleDecember 2015
Revisiting Asynchronous Linear Solvers: Provable Convergence Rate through Randomization
Journal of the ACM (JACM), Volume 62, Issue 6Article No.: 51, Pages 1–27https://doi.org/10.1145/2814566Asynchronous methods for solving systems of linear equations have been researched since Chazan and Miranker's [1969] pioneering paper on chaotic relaxation. The underlying idea of asynchronous methods is to avoid processor idle time by allowing the ...
- research-articleNovember 2015
Regular Languages Are Church-Rosser Congruential
Journal of the ACM (JACM), Volume 62, Issue 5Article No.: 39, Pages 1–20https://doi.org/10.1145/2808227This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of ...
- research-articleSeptember 2015
Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata
Journal of the ACM (JACM), Volume 62, Issue 4Article No.: 30, Pages 1–33https://doi.org/10.1145/2789208A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multitype branching processes, stochastic context-free grammars, probabilistic ...