# Questions tagged [lower-bounds]

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

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### Confusion about lower bounds and upper bounds in learning theory

In computer science, lower bounds and upper bounds are defined as follow: $$m \geq g(n) \implies m = \Omega(g(n))$$ $$m \leq g(n) \implies m = \mathcal{O}(g(n))$$ However, in proving lower bounds and ...
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### 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)...
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### 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 ...
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### 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 ...
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### Parity and $AC^0$

Parity and $AC^0$ are like inseparable twins. Or so it has seemed for the last 30 years. In the light of Ryan's result, there will be renewed interest in the small classes. Furst Saxe Sipser to Yao ...
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### Examples of the price of abstraction?

Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely: It is known that Gaussian ...
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### 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 ...
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### Reference to lower bound on separator in a grid?

It is easy to verify that given the d dimensional grid of the integer points $\{1,\ldots,n\}^d$, with the regular adjacency, one can find a separator of size $n^{d-1}$ (just pick any middle hyperplane,...
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### 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 ...
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### Optimal greedy algorithms for NP-hard problems

Greed, for lack of a better word, is good. One of the first algorithmic paradigms taught in introductory algorithms course is the greedy approach. Greedy approach results in simple and intuitive ...
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### Deciding if all matrix multiplication entries have at least two witnesses

Consider two square matrices $A(x,y)$ and $B(y,z)$ of dimensions $N×N$ containing boolean entries. Consider the output product matrix $C(x,z)$ where $C=AB$ (not boolean matrix multiplication but the ...
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### 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 ...
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### How to prove that USTCONN requires logarithmic space?

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold ...
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### Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?

Background It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$. Although some natural problems are known to exist, many of them ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### Lower bounds on #SAT?

The problem #SAT is the canonical #P-complete problem. It's a function problem rather than a decision problem. It asks, given a boolean formula $F$ in propositional logic, how many satisfying ...