# Questions tagged [np]

In computational complexity, NP is the complexity class consisting of problems whose yes instances can be verified in polynomial time. NP stands for 'nondeterministic polynomial time '.

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### If NP=P, then does quantum mechanics have a classical interpretation? [closed]

Is it true that $BQP\subset NP$, if $NP=P$?
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### Root of polynomials in a finite field

I am looking for a way to find out if a polynomial $P\in \mathbb Z/p\mathbb Z=\mathbb F_p$, of great degree, has roots in $\mathbb F_p$, with $p$ a big prime number. For example : $p=2^{2020}-69$ ...
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### Techniques for proving relaxed one-wayness of functions

Existence of one-way functions is a widely accepted conjecture in complexity theory. A function is one-way if it is computable in polynomial-time but not invertible in polynomial-time (this is ...
704 views

### Representing mathematical statements as SAT instances

The following problem (call it THEOREMS) belongs to class NP. Input: Mathematical statement $S$ (written in some formal system such as ZFC) and positive integer $n$ written in unary. Output: "Yes" if ...
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### (How) do Better TSP Heuristics help in Answering the $NP=P$ Question?

This question is motivated by my impression, that finding better heuristics for the TSP problem (or any other $NP$-complete problem) is "only" of practical interest, but doesn't provide any progress ...
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### NP-hardness of finding 0-1 vector to maximize rows of {-1, +1} matrix

Consider the following discrete optimization problem: given a collection of $m$-dimensional vectors $\{ v_1, \dots, v_n \}$ with entries in $\{-1, +1\}$, find an $m$-dimensional vector $x$ with ...
1 vote
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### Cost associated set problem NP-hard

I have the following problem. I wonder whether or not it appears in the literature. Is it NP-hard? Given a set $S = \{1,2,\ldots,m\}$, and $A_1,\ldots, A_n$ are subsets of $S$. Each set $A_i$ has ...
268 views

### Finding a subgraph of cliques with the minimum total sum weight

Consider the following graph problem. For a number $K$ and a set $\mathcal{K} = \{ 1, \ldots,K\}$, we have a set of vertices $V_k^s$ for all $s \subset \mathcal{K} \setminus \{k\}$, $s$ is not empty ...
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### Simple cake cutting puzzle

I got interested in a cake cutting problem from computational perspective. Suppose we have a piece of cake and we want to slice it into pieces using several cuts (straight lines). Each cut represents ...
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### Is there a polynomial-time algorithm for untangling the unknot?

I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (...
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### Continuous backpack with multiple choice items. NP prove

Continuous backpack with multiple choice items is the problem where you need to collect items by one from each of distinct sets and associate them with rational numbers so that their sum of weights ...
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### What arguments do exist against defining completeness in NP using injective Karp reductions?

It is crucial to use the right notion of reduction to define completeness inside NP. Different notions of completeness inside NP may have significant impact on the properties of complete languages. ...
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### How slow are direct solutions of NP-complete problems on computers?

Sometimes I see that people call a problem NP-hard and because of that refuse to create computer algorithms that directly solve it. I think I've never read actual benchmark results for such problems. ...
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### Covering a graph by trees with depth constraint

Given a graph $G$ and a depth constraint $h$, my question is: what is the complexity to find a tree cover of $G$, denoted as $T=\{T_1, T_2, ..., T_n\}$. For each $T_i$, its depth(height) is no larger ...
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### Minimum cover for sets in which each element appears in exactly 2 sets?

Is there an algorithm for finding minimal covers of a set of sets in which each element of the universe appears in exactly 2 sets? I realize that LP relaxation approximates this to within a factor of ...
1 vote
### How I can prove the equality $P^{P_{\operatorname{space}}}=NP^{P_{\operatorname{space}}}=P_{\operatorname{space}}^{P_{\operatorname{space}}}$ [closed]
I know how to prove that if $A \in P^{P_{\operatorname{space}}}$ then $A \in NP^{P_{\operatorname{space}}}$ and $A \in P_{\operatorname{space}}^{P_{\operatorname{space}}}$. I don't know how to prove ...
I am interested in the complexity of the following problem: GENERAL-SQUARE-PRODUCT INSTANCE: Two sets $A=\{a_1,\ldots,a_n\}$ and $B=\{b_1,\ldots,b_n\}$ of integers, a positive integer $k<n$ and a ...