# Questions tagged [lower-bounds]

questions about lowerbounds on functions, usually the complexity of an algorithm or a problem

279 questions
Filter by
Sorted by
Tagged with
1k views

161 views

### Are exponential lower bounds known against $MOD_6 \circ MOD_3$ circuits computing $OR$?

Background What is currently known for depth-2 $CC^0$ circuits with restricted gate types: In  it is shown that $(MOD_p)^k \circ MOD_m$ circuits (that is, $k$ layers of $MOD_p$ gates at the output)...
134 views

### Counting argument for LTF circuits

In Boolean circuit complexity, Shanon's counting argument shows that a random Boolean function on $n$-input bits requires a circuit of size $\Omega(2^n/n)$ to be computed by a circuit made of AND, OR ...
372 views

### What is (a reasonable conjectured lower bound on) the query complexity of solving an $n\times n$ system of linear equations given space $O(n)$?

I am faced with the following problem: A uniformly random $n \times n$ matrix $M$ over a finite field $\mathbb{F}$ is sampled. The algorithm has oracle access to the matrix entries, and each query to ...
143 views

### Time/space lower bounds on Majority (in the multitape TM model)

MAJORITY is the language of bitstrings where more than half of the bits are 1s. I'm interested in lower bounds in the multitape TM model. This can be solved in $DTISP(O(n), O(\log(n))$ with a naive ...
188 views

### Exact lower bound on matrix multiplication

The recent publication in Nature of "Discovering faster matrix multiplication algorithms with reinforcement learning" by Fawzi et al. has shown a method for discovering fewer element ...
679 views

171 views

### Triangle detection hardness in regular graphs

Consider a tripartite graph over $n^{1-\epsilon}$ vertices each in sets $I, J, K$. Suppose we impose a constraint that every vertex has degree $n^\epsilon/c$ for some constant $\epsilon > 0$ and ...
78 views

### Obtaining a lower bound of a matrix norm

I was wondering (on a setting where $\vec X_i \sim \mathcal{N}(\vec\mu, \mathbb{I})$ are $n$ random $d$-dimensional multivariate normal vectors with unknown mean $\vec\mu$) how I could obtain a lower ...
59 views

### What is known about the stabilizer rank of this simple state?

Consider the uniform superposition of all length-$n$ bit-strings of Hammming weight $w$, $$|\phi_w\rangle =\sum_{x\in \{0,1\}^n,|x|=w} |x\rangle$$ What is known or conjectured about the stabilizer ...
343 views

### ETH based lower bound for $k$-COLORING of bounded degree graph

It is known that there is no $2^{o(n)}$-time algorithm for 3-COLORABILITY of graphs of maximum degree four, unless ETH fails . Is a there a similar result for $k$-COLORABILITY assuming only ETH (...
135 views

### What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
181 views