# Tag Info

### 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 ...
Accepted

### Matrix permanent is 0

Expanding my comment: Computing the permanent of a matrix is #P-hard (Valiant 1979) even if the matrix entries are all either 0 or 1. We can interpret a matrix $M \in \{0,1\}^{n \times n}$ as a ...

### 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 ...
Accepted

### Complexity of k-clique for hypergraphs

It is not known if there is an $\varepsilon > 0$, $c > 2$, and $k > c$ such that $(c,k)$ hyperclique is in $n^{k-\varepsilon}$ time. Note that the case of $k \leq c$ is trivial. For years I ...

### 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 ...

### Approximating the sign rank of a matrix

Recent work by Alon, Moran, and Yehudayoff gives an $O(n/\log n)$ approximation algorithm. Let $d$ be the VC-dimension of a sign matrix $S$. The idea is that there exists an efficiently computable ...
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 ...
Accepted

### Matrix vector multiplication algorithm using minimal number of additions

This is essentially the problem that motivated Valiant to introduce matrix rigidity into complexity (as far as I understand the history). A linear circuit is an algebraic circuit whose only gates ...
Accepted

### Exact formula for the number of spanning trees of a rectangle

According to https://www.cse.ust.hk/~golin/pubs/ANALCO_05.pdf there is no closed-form formula known. According to http://arxiv.org/pdf/cond-mat/0004341v1.pdf the number is asymptotic (for $n$ and $m$ ...