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Questions tagged [open-problem]

Problems without an answer after substantial efforts by many users. This tag indicates the problem is of graduate research level and beyond. Do not use this tag for a homework question. Do not use this tag for a question that do not have a solution that can be proved mathematically.

In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved. This explanation comes from Wikipedia.

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Is the language of pairs of words of equal length whose hamming distance is 2 or greater context-free?

Is the following language context free? $$L = \{ uxvy \mid u,v,x,y \in \{ 0,1 \}^+, |u| = |v|, u \neq v, |x| = |y|, x \neq y\} $$ As pointed out by sdcvvc, a word in this language can also be described as the concatenation of two words of the same…
Robert777
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Flow graph that requires pushing back flow in Ford Fulkerson

Does there exist a flow graph that always requires flow to be pushed back no matter what ordering of augmenting paths is chosen in Ford Fulkerson? Let's assume we use the standard procedure of repeating this step: Find an augmenting path $p$ in…
ryan
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Worst-case input for median-of-medians with groups of size 3

Typically, median of medians algorithm is written with groups of size $5$ or $7$ to ensure worst-case linear performance. The argument against groups of size $k=3$ is typically that we get a recurrence of: $$\begin{align*} T(n) & \leq T(n/3) +…
ryan
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Conjecture: a half of a pairing context-free language must be a regular language

If $A$ and $B$ are languages, let $A\bowtie B$ denote the set of strings made by concatenating any word from $A$ and any word from $B$ of equal length. $$A\bowtie B \equiv \{ ab : a\in A,\;b\in B, |a|=|b|\}.$$ You can prove using regular grammars…
user326210
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Is the smallest grammar problem over the singleton alphabet known to be NP-complete or ...?

I can't find straight answer on this via googling around. Can you provide a reference? The smallest grammar problem over a singleton alphabet is to find the smallest CFG $g$ that produces one and only one string of $a$'s, i.e. $s = a^n$ for some…
Daniel Donnelly
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Topological sort where some nodes can't come in between two other nodes

I have a DAG which I would like to do a topological sort on but there is a catch. I also have a relation NotBetween(X,Y,Z) which means that in the sort the node Y cant come "in between" node X or node Y. In other words, Y < X OR Z < Y. I've thought…
Isak
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Is the buckets of water problem in NP?

Continuing from this question: the buckets of water problem. (All the definitions can be found there, so I will not repeat them). As seen there by Yuval's answer, the problem is NP-Hard. I was attempting to prove its NP-completeness, and while doing…
Eric_
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Changing a matrix to become an ancestry matrix

An ancestry matrix $M$ for rooted tree $T$ is defined as $M[ij] = 1$ iff node $i$ is an ancestor of node $j$. Suppose we are given a matrix $X$. We can easily check that if $X$ is compatible with some rooted tree (and create the tree) or not. The…
Dandelion
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$\textbf{NP}\neq \textbf{DTIME}(2^{\sqrt{n}})$

I want to prove that $\textbf{NP}\neq \textbf{DTIME}(2^{\sqrt{n}}).$ My thoughts is: if I try to prove $\textbf{NP}\not\subseteq \textbf{DTIME}(2^{\sqrt{n}})$ would imply $\textbf{NP $\neq$ P}$. if I try to prove…
Xoxoxo
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I would like to know what are the directions to work on if I want to prove that $NP=coNP$?

I am currently learning about NP and coNP related content and have been exposed to the$NP \overset{\text{?}}{=}coNP$ problem. I would like to know what are the directions to work on if I want to prove that $NP=coNP$? For example, currently I know…
lz9866
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Determine whether we can pack $(r_1, \dots, r_k)$ copies of the variables $(A_1, \dots, A_k)$ into a string $s$, in time polynomial in $|s|$?

Let $s \in \Sigma^*$ be a finite string over alphabet $\Sigma$. Find all "compressible" substrings of $s$, that is substrings $t \leqslant s$ (notation) such that $t\alpha t \leqslant s$ for some $\alpha \leqslant s$ and $|t| \geq 2$. It's true…
Daniel Donnelly
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What are open problems in computer science?

I should prepare some paper for a colloquium (kinda student-task) and it should cover the following points: (1) at least one notable discovery in theoretical informatics (or computer science) (2) at least one open problems in theoretical…
Constantin
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Open Problem: Structural Learnability of Pseudo-Random Boolean Circuits

I would like to propose an open problem at the intersection of computational complexity, pseudorandomness, and circuit theory. This problem has potential implications for cryptography, AI model analysis, and the theory of explainability in…
Konstant
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( Soft question ) P vs NP - is such a situation possible?

Currently P vs NP is the holy grail of theoretical computer science. And the nature of the problem is as such that if actually P = NP is proved then most of the proofs for mathematical statements would be trivial to find, and on the exact other end…
Aditya Mishra
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Can we simply consider a pseudo random number generator to be a function $f: \Bbb{Z}_n \to \Bbb{Z}_n$ for ever-increasing $n$?

On modern architectures, random number generators get seeded by the current system time as a source of randomness, which is nice because it is kind of unpredictable when a process will switch to the current process. When this timer is in…
Daniel Donnelly
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