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The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems…
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced…
The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input…
A tight lower bound for required I/O when computing an ordinary matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of segments needed…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…
Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix multiplication, must be ordinary matrix…
In this paper, we study the $d$-dimensional update-query problem. We provide lower bounds on update and query running times, assuming a long-standing conjecture on min-plus matrix multiplication, as well as algorithms that are close to the…
One fundamental question in database theory is the following: Given a Boolean conjunctive query Q, what is the best complexity for computing the answer to Q in terms of the input database size N? When restricted to the class of…
Min-plus matrix multiplication is used in many problems operating on distances in graphs or solvable by dynamic programming. Assuming the APSP hypothesis, there is no subcubic-time algorithm for the min-plus product of two general $n\times…
Obeying constraints imposed by classical physics, we give optimal fine-grained algorithms for matrix multiplication and problems involving graphs and mazes, where all calculations are done in 3-dimensional space. We assume that whatever the…
In this paper, we present novel deterministic algorithms for multiplying two $n \times n$ matrices approximately. Given two matrices $A,B$ we return a matrix $C'$ which is an \emph{approximation} to $C = AB$. We consider the notion of…
Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying…
Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the $\mathcal{O}(n^3)$ time algorithm…
We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three $n \times n$ matrices $A$, $B$, and $C$ as input, to decide whether $AB = C$. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV…
Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction…