32 votes

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for ...
Josh Alman's user avatar
16 votes
Accepted

Memory requirement for fast matrix multiplication

The space usage is at most $O(n^2)$ for all Strassen-like algorithms (i.e. those based on upper bounding the rank of matrix multiplication algebraically). See Space complexity of Coppersmith–Winograd ...
Ryan Williams's user avatar
15 votes

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

Well, one thing is I think that all the constructions we know of - and even the families of potential constructions that people have proposed (e.g., Cohn-Umans approaches, generalizations of ...
Joshua Grochow's user avatar
13 votes
Accepted

What are the consequences of solving XOR 3-SAT in Logspace?

Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your ...
Gustav Nordh's user avatar
  • 1,047
12 votes
Accepted

Checking equivalence of two polytopes

I cannot say for sure if you will consider the following approach as better, but from a complexity-theoretic point of view there is a more efficient solution. The idea is to rephrase your question in ...
Christoph Haase's user avatar
12 votes

Finding the sparsest solution to a system of linear equations

Consider the problem $\text{MAX-LIN}(R)$ of maximizing the number of satisfied linear equations over some ring $R$, which is often NP-hard, for example in the case $R=\mathbb{Z}$ Take an instance of ...
Joe Bebel's user avatar
  • 2,295
12 votes

Estimating the rank of a large sparse matrix

There is a very neat randomized algorithm by Cheung, Kwok, and Lau, which computes the rank of an $n\times m$ matrix $A$, $n\le m$, in time $O(\mathrm{nnz}(A) + \min\{n^\omega, n\cdot\mathrm{nnz}(A)\})...
Sasho Nikolov's user avatar
11 votes

Status of Raghavendra's algorithm for solving linear systems in finite fields

This is a paper that seems to improve upon the result. https://arxiv.org/pdf/1209.3995.pdf
user43170's user avatar
  • 163
10 votes
Accepted

The minimum number of arithmetic operations to compute the determinant

It is known that the number of arithmetic operations necessary to compute the determinant of an $n\times n$ matrix is $n^{\omega+o(1)}$, where $\omega$ is the matrix multiplication constant. See for ...
A. Rex's user avatar
  • 266
10 votes
Accepted

The complexity of the permanent of low rank matrices

The paper "Two algorithmic results for the traveling salesman problem" by Barvinok describes an $n^{\mathcal{O}(m)}$ algorithm for computing the permanent of a rank-m matrix. I don't know whether this ...
smapers's user avatar
  • 839
9 votes

Non-Orthogonal Vectors Problem

When $k$ is given as part of the input, the second problem is equivalent to the monochromatic Max-IP problem (given $S \subseteq \{0,1\}^d$, find $\max_{(a,b) \in S, a\ne b} a \cdot b$). Recently I ...
Lijie Chen's user avatar
8 votes
Accepted

Matrix multiplication with transpose

No. Consider block matrices $A = \left(\begin{matrix} 0 & 0 \\ 0 & X \end{matrix}\right)$ (with symmetric $X$) and $B = \left(\begin{matrix} 0 & Y \\ 0 & 0 \end{matrix} \right)$. ...
Vladimir Lysikov's user avatar
8 votes
Accepted

Status of Raghavendra's algorithm for solving linear systems in finite fields

The paper by Raghavendra is now also published and available here under the title: Correlation Decay and Tractability of CSPs, appeared in the 43rd International Colloquium on Automata, Languages, ...
LeoW.'s user avatar
  • 118
8 votes
Accepted

Binary vector $t$ in $span(S)$ over $\mathbb{Z}/q\mathbb{Z}$ for all prime powers $q$ $\Rightarrow$ $t$ in $span(S)$ over $\mathbb{Z}$?

The revised conjecture is true, even under relaxed constraints on $S$ and $t$—they may be arbitrary integer vectors (as long as the set $S$ is finite). Notice that if we arrange the vectors from $S$ ...
Emil Jeřábek's user avatar
7 votes

Number of solutions for a system of linear equations over a finite ring

The answer to (1) is yes (regardless of the properties D.W. asked for in the comments), depending on how $R$ is given: First, note that since $R$ is finite, the abelian group $(R,+)$ is of the form $\...
Joshua Grochow's user avatar
7 votes

Non-Orthogonal Vectors Problem

If $k=O(\log n)$ I believe the techniques of Alman, Chan, and Williams give the best known solution to the Non-Orthogonal Vectors Problem. (They phrase it differently, as a Hamming closest pair ...
Rasmus Pagh's user avatar
6 votes
Accepted

Finding the number of independent rows of a matrix

(what follows solves the question for columns, easily adapted to rows either by transposing or by doing a column echelon instead of a row one) You can do the following: put your matrix in reduced ...
Marc's user avatar
  • 686
6 votes
Accepted

Min Hamming distance of a given string from substrings of another string

Elaborating Paul's suggestion for a $O(n \log n)$-time algorithm: Input: Let $u \in [m]^k$ and $v \in [m]^n$ with $k \leq n$, where $U=[m]=\{1,2,\cdots,m\}$. Define polynomials $$p(x,y) = \sum_{i \...
Thomas's user avatar
  • 2,803
6 votes

Finding the sparsest solution to a system of linear equations

The problem is NP-complete, by reduction from the following problem: Given an $m\times n$ matrix $A$ with integer entries and an integer vector $b$ with $n$ entries, does there exist a 0-1 vector $x$ ...
Gamow's user avatar
  • 5,772
6 votes
Accepted

Graph isomorphism problem with invertible adjacency matrices

Isn't this graph just a collection of cycles? So we only need to compare that all the lengths in the two graphs match (which can be done by sorting the lengths).
Igor Shinkar's user avatar
  • 1,897
6 votes
Accepted

Dichotomy of the spectra of directed graphs

The answer to “alternatively, can a directed graph have an eigenvalue with an exponentially small imaginary component” is YES (though I don’t understand what is “alternative” about this statement, as ...
Emil Jeřábek's user avatar
6 votes

How many multiplications are needed to compute the determinant of a 3×3 matrix?

I have a partial answer to this now. I still don’t know whether anyone had ever explicitly written down a method for computing the 3×3 determinant using 8 multiplications, but anyone with sufficient ...
Robin Houston's user avatar
6 votes
Accepted

Strongly polynomial time algorithm for shortest convex combination

It is known via a paper of De Loera, Haddock and Rademacher that a strongly polynomial time algorithm for finding a minimum norm point in a simplex implies a strongly polynomial time algorithm for ...
Chandra Chekuri's user avatar
5 votes
Accepted

Estimating the rank of a large sparse matrix

This has also been studied from the property testing / query complexity point of view. For example: Li, Wang, & Woodruff. Improved Testing of Low Rank Matrices, KDD '14. (Freely available ...
Joshua Grochow's user avatar
5 votes

Complexity of Finding the Eigendecomposition of a *Symmetric* Matrix

I know this is a really old question, but it seems like this recent paper https://arxiv.org/abs/1912.08805 improves the runtime to $O(n^\omega)$, down from $O(n^3)$.
walkerbacker's user avatar
5 votes

Is there a polynomial time algorithm to determine if the span of a set of matrices contains a permutation matrix?

Theorem. The problem in the post is NP-hard, by reduction from Subset-Sum. Of course it follows that the problem is unlikely to have a poly-time algorithm as requested by op. Here is the intuition. ...
Neal Young's user avatar
  • 9,595
5 votes
Accepted

Has anyone mixed linear algebra with formal language theory in this way?

How you choose your vector $\nu$ for every terminal symbol you must have a row with exactly one $\epsilon$ in your matrix so that it is a fixed point. So we could disregard terminal symbols, and what ...
StefanH's user avatar
  • 2,037
5 votes

Checking equivalence of two polytopes

The fact that the underlying polytope $Ax \le b$ is the same for $P_1$ and $P_2$ does not matter, unless we know something specific about $A$ and $b$. This is because a general polytope is an affine ...
Denis Pankratov's user avatar
5 votes

Algebraic account of Gaussian elimination?

You really should read Gowers' essay carefully - it cleanly details the reasons why you need a basis in general. So if there is going to be an algebraic account of Gauss-Jordan, it will necessarily ...
Jacques Carette's user avatar

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