# 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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### Shortest Lattice Vector with restricted $x$

Let $\Lambda$ be a lattice with basis, $B$ consisting of vectors $b_i$, so that the elements of $\Lambda$ are of form, $y\in \Lambda \iff y=Bx=\sum_i b_ix_i$ for some $x_i\in\mathbb{Z}$.
My questions ...

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### Is it known whether $\mathrm{NP \subseteq P/poly}$?

It is not immediately clear to me whether this statement is true or false. Can finite restrictions of NP problems be computed in polynomial time?

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### Karp hardness of two cycles which lengths differ by one

Our problem is as follows:
NEARLY-EQUAL-CYCLE-PAIR
Input: An undirected graph $G(V,E)$
Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO
Is ...

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### $\mathrm{NP}$-complete problems in graph theory: undirected vs. directed

Is it true that it is much easier to establish $\mathrm{NP}$-complete on undirected graphs than digraphs (directed graph)?
Academic articles proving $\mathrm{NP}$-completeness of problems on ...

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### descriptive complexity theory to attack computational complexity problems [closed]

What is the usefulness of descriptive complexity to attack computational complexity theory?what are the recent results in this direction? Thanks

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### Bipartite clustering is NP-hard?

Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, ...

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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 ...

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### 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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### Proving a Property of Crossing Generalized Diagonals

I am trying to prove the graph-theoretic generalization of the following geometric observation:
let $\mathbb{P}$ be a finite set of distinct points in the Euclidean plane and $card\left(CH\left(\...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### How to complete the NP-hardness proof of GENERAL-SQUARE-PRODUCT?

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 ...

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### Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times 1}$, ...

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### The complexity of Max-K interval selection

I came up with the following problem, but do not know how to analyze it.
Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in $...

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### Is finding a local minimizer of a NP-hard optimization problem is still NP-hard [closed]

I was wondering if for a NP-hard optimization problem, I only want to find its local minimizer, is it still NP-hard or NP-hard is only true when trying to find a global minimizer?

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### Can we say that this problem is NP-hard?

I have an optimization problem of the form:
\begin{align}
&\text{maximize}\quad f(\mathbf{x}) = \dfrac{\sum\limits_{n=1}^{N}x_na_n}{1+\sum\limits_{n=1}^{N}x_nb_n}\\
& \text{subject to}\quad \...

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### On Knot Equivalence problem statement

How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...

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### How hard is recognizing a permutation that is a square for the shift product?

This is a continuation of my attempts to generate simple combinatorial computational problems that turn out to be computationally hard (NP-complete). In this pursuit, I came up with a permutation ...

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### NP-hardness of finding maximum of minimum element in diagonal of a matrix

For $A = \{a_{ij}\} \in R^{n\times n}$, is finding
$$
\max_{\sigma \in S_n}\min_{1 \le i \le n} a_{i,\ \sigma_i}
$$
NP-hard?

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### Surd Partition Problem

Could the following "Surd Partition" problem be NP complete? Note that if the square roots are omitted in the following then the problem is well known to have a polynomial solution.
Surd Partition
...

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### Is this problem on weighted bipartite graph solvable in polynomial time or it is NP-Complete

I encounter this problem recently and I want to know whether it is NP-Complete or solvable in polynomial time:
Given a undirected weighted bipartite graph $G = (V, E)$ where $V$ can be partitioned ...

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### reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...

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### What impact would P=BQP have on NP?

Assuming P=BQP (ie we have polynomial time algorithms to solve all BQP problems) can we use it to prove that P=NP?
The argument is that since we have the Grover's algorithm which can solve NP ...

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### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...

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### Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ ...

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### Is the domination number NP for non-bipartite graphs?

Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?

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### Simple example of why Differential Equations can be NP Hard [closed]

Just looking for a simple example of why Differential Equations can be NP hard
Edit:
It appears that the answer below may be what I was looking for, but I am clarifying just in case:
Slides 58--...

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### Conjecture on NP-completeness of tesselation of Wang Tile up to finite size

Motivated by these following questions on tessellation:
coloring in lattice
Reference for Wang Tile
Computational approach deciding whether a set of Wang Tile could tile the space up to some size
...

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### How to determine the distance between two matrices under the meaning of a matrix function? [closed]

Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a ...

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### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord (...

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### Reference Request for: Finding Large Bipartite Subgraphs via Destruction of Odd Cycles in Graphs

From the observation, that a bipartite graph doesn't contain odd cycles, it would seem natural to attempt to destroy all odd cycles in the most efficient way, by either removing edges or vertices of ...

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### How hard is reconstructing a permutation from its differences sequence?

My interest in combinatorially motivated computational problems led me to search for simple problems that turn out to be computationally hard. In this pursuit, I came up with a problem which I hope is ...

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### Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition problem is strongly NP-...

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### Testing contrasts in statistics: Is this provably a hard problem, or not?

Scheffé's method for identifying statistically significant contrasts is widely known. A contrast among the means $\mu_i$, $i=1,\ldots,r$ of $r$ populations is a linear combination $\sum_{i=1}^r c_i \...

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### Strategic vertex labeling

We are given a graph $G=(V,E)$ with positive edge weights $w_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0(all vertices with ...

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### Are set covering problems with nonlinear cost functions NP-Hard?

Are set covering problems (set cover problem wikipedia) with a nonlinear cost function also NP-hard? Is there a general result about this?
To be more specific the cost function I am interested in ...

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### Generating 3SAT circuit for Integer factorization example

I read somewhere that 3SAT can be used to solve Integer Factorization.
If that is true, could someone teach me a simple example of generating the 3SAT by using a small number? Let's say you are given ...

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### Non-trivial lower bound on the number of “Graph Diagonals”

The definition of Graph Diagonals, that are the subject of this question, is based on the notions of crossing edges and on connected graphs:
Two edges $AC$ and $BD$ of a complete, symmetric and ...

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### Are there sampNP-intermediate problems?

This questions is approximately cross-posted from theoretical computer science stackexchange
Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\mathsf{NPI} := \mathsf{NP} \...

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### SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...