Questions tagged [np]

Questions about decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input.

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Implications in $\mathbb{NP}$-completeness

Suppose I have an $\mathbb{NP}$-Complete problem called problem $A$. Further, suppose that $A$ is poly-time solvable in undirected-acyclic graphs; in other words, trees. Now, If I take a problem ...
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Finding the common part of two systems of linear equations modulo 2

Given two systems of linear equations modulo 2, is there a method to find the common part (the equations which are true in both case) ? For example... Matrix 1: a⊕b=1 and c⊕d=0 and y=1 and e=0 Matrix ...
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Is NP in NP/Poly?

The answer is yes, NP/poly is defined as the class of problems solvable in polynomial time by a non-deterministic Turing machine that has access to a polynomial-bounded advice function--the advice ...
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How can I create the table and solve the error, why did the error happened? [closed]

import world_population.csv population = Table.with_columns( "Population", population_amounts, "Year", years ) population No module named 'world_population'
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proving that a problem is in P

I read online that this problem is in P: Problem = {a^n, where n is a primary number} I can't find any algorithm that decides if a word w in ...
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Proving NP-completeness for a not so cheesy problem

Let's say we have a matrix M[1..B, 1..B] (i.e., a square matrix) and a mouse in the upper left corner (1,1). We also have an integer A, which tells how many pieces of cheese there are in the matrix. ...
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Why do I only need to reduce from one problem in NP to prove NP-Hardness?

Suppose I wish to show that my decision problem $Q$ is NP-Hard. Why do I need to reduce from one problem $Q'$ of known hardness ? Consider for instance the following situation: Here, I have my set NP ...
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Would $\mathsf{P=BPP}$ imply $\mathsf{dIP=IP}$ and if not then why?

Complexity class $\mathsf{IP}$ includes all problems that can be solved using an interactive proof system where the verifier is a probabilistic polynomial time machine, and the prover is a machine of ...
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If A is not in NP, and A reduces to B, does this mean B is not in NP?

I know it is true that if A is not in P, and A reduces B, then B is not in P. But is it true for NP as well? If A is not in NP, and A reduces to B, does this mean B is not in NP? Why or why not? ...
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How to Show Subset Sum $\le_p$ 3-Partition

Given a set of integers S (positive and negative, may contain duplicates) can S be divided into three disjoint subsets that all sum to the same value? Prove this problem is NP-complete. Is it ...
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Showing that Subset Sum Reduces to 3-Partition [duplicate]

Given a set of integers S (positive and negative, may contain duplicates) can S be divided into three disjoint subsets that all sum to the same value? Prove this problem is NP-complete. This is a ...
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For every $\mathrm{NP}$ language $L$, is there a verifier such that, for all the certificates $u$ of other verifiers of $L$, it accepts $(x, u)$?

Let $L$ be an $\mathrm{NP}$ language. Then there exists a verifier $V$ of $L$ and a polynomial $p\colon \mathbb{N} \to \mathbb{N}$, such that for every $x \in \Sigma^{*}$, $x \in L$ if and only if ...
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Proving the language of non-primes is in NP

I am learning about NP problems and found this problem in my textbook that I was not sure how to answer, and was looking for some help on how to start the question. Show the following language is in ...
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Are all reductions from NP-complete problems either NP-complete or are contained in P?

Let's say we have a problem $A \in \mathsf{NP}$. Now let's say we have a reduction $f(\mathsf{SAT}): A \leq \mathsf {SAT}$. So, assuming that $A$ is not $\mathsf{NP}$-complete we have that $f(\mathsf{...
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Is there a decision problem in NP whose corresponding function problem is not in #P?

I am trying to get an imagination of the class #P for my bachelor thesis. Right now I think of it as a DTM that runs every possible path to run an algorithm on some decision problem at once. But in ...
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NP-completeness of testing whether a SAT formula does not contain any redundant clause

This paper explains that testing the irredundancy of a SAT instance is NP-complete. But I don't understand the theorem/reduction. How would one reduce 3SAT or k-SAT to this problem, for example?
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What is a good reference for NP hardness in the machine learning/optimization/operations research context?

I am reading some papers in machine learning and at the very beginning (introduction) you will see statements of theorems that says, for example: Theorem 1.1. For any constant ϵ > 0, it is NP-hard ...
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If a poly-time solution exists for an NP-Complete (Decision) problem, then there exists a poly-time solution for the NP-hard (Optimization) flavor?

This question boils down to: If a polynomial time solution exists for a decision problem, is there also a polynomial time solution for the same problems optimization flavor? Let's take the Traveling ...
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Why does SAT-UNSAT $\in NP \implies NP = coNP$

I was reading this post about the DP completeness of the problem SAT-UNSAT (both are well defined in this post). The answer added a note at the end that states the class complexity DP differs from NP, ...
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Is it a sufficient condition to be in NP?

Suppose the following situation. You have a decision problem $D$. You know that $SAT$ is $NP$-complete. You know that $D\leq_p SAT$. Can you conclude that $D\in NP$? I think it's true because it ...
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Can quantum computing help solve NP-Complete problems?

i was just wondering if quantum computing has done any good so far in solving NP complete problems. I am aware that quantum computing does solve some NP problems which are classically hard in ...
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P vs NP characterization confusion

I know that $P \subseteq NP$, but for a problem in $P$, e.g. MST in a graph, is it a correct statement to say that: The MST problem belongs in NP-Class. (I mean, i think it is correct, but could ...
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Need help to proof that NP⊆NP4

NP4 = { L | There exists a non deterministic polynomial Turing machine M, such that for every x∈L, M accepts x on at least 4 paths in the computational tree of M on x. and for every x∉L,M accepts x on ...
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algorithm for checking satisfiability

In order to prove that SAT is in NP, I need to come up with a polynomial time verfier (an algorithm). The Cooks Levin Theorem uses a non-deterministic Turing machine but that's not what I am looking ...
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are all problems in P in NP [duplicate]

I know this is a similar question to this post, but I want to further clarify my understanding. In the picture from wikipedia: I understand that every problem that's in $P$ is also in NP. does this ...
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Is it incorrect too say that this function problem cannot be in $FNP$?

Decision Problem: Is $2^k$ + $M$ NOT a prime? $K$ and $M$ are our inputs represented as integers. Function Variant: Output the result of $2^k$ + $m$ We can consider, $M$ = $0$. Proof that calculating ...
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Can an NP-Complete problem be reduced to an NP problem?

All NP problems can be reduced to NP-Complete problems, can an NP-Complete problem be reduced to a NP problem (non complete)?
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Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
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$NP$ is not in $P(n^k)$ for any fixed $k \geq 1$

I encountered this problem which asks to show that for any fixed $k \geq 1$, $NP$ is not contained in $P(n^k)$... As an attempt, I thought of using the time hierarchy theorem which says that there ...
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I can verify solutions to my problem in polynomial time, how would a non-deterministic algorithm arrive to a solution if it always takes $2^n$ bits?

Decision Problem: Given integers as inputs for $K$ and $M$. Is the sum of $2^k$ + $M$ a $prime$? Verifier ...
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Nondeterministic polynomial time algorithm versus certificate/verifier for showing membership in NP

In this paper (https://arxiv.org/pdf/1706.06708.pdf) the authors prove that optimally solving the $n\times n\times n$ Rubik's Cube is an NP-complete problem. In the process, they must show that the ...
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Does $L \leq_p K$ and $K\in NP$ implies $L \in NP$?

I only find theorems like $L \leq_p K$ and $K\in P$ implies $L \in P$ in books, but I think the same should be true for NP as well. Or is there anything I am missing? I am asking since this would be ...
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Why isn't SAT in coNP?

I understand why NP=coNP if SAT is in coNP (How do I prove that SAT in coNP implies NP=coNP?). But I'm missing why the following machine doesn't turing recognize the complementary of SAT: Given a ...
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MAXSAT using dpll algorithm?

It's possible to return from a dpll algorithm M as maximum for MAX-SAT problem?: I have a sample: https://gist.github.com/davefernig/e670bda722d558817f2ba0e90ebce66f we can modify recurrency to return ...
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Proof of Co-Problem being in NP if Problem is in NP using negated output

Given any problem $P$ that we know of being in $NP-\text{complete}$, where is the flaw in the following proof? Given a problem $Co-P$ which is the co-problem of $P \in NP-\text{complete}$, $Co-P$ is ...
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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 ...
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Proof for NP-hardness of simultaneous minimization and maximization of a weighted subset

I am working on a problem defined as the following Given a set of $n$ elements called $R \subseteq \mathbb{N} \times \mathbb{N}$ and numbers $Z,G \in \mathbb{N}$, where $Z$ is a measure of our ...
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Does P not NP imply NP COMPLETE disjoint from RP?

According to Wikipedia https://en.wikipedia.org/wiki/RP_(complexity), $P \ne NP$ implies that $RP$ is a strict subset of $NP$. Does anybody have a reference? Furthermore, am I correct that if this ...
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complexity of dividing set of number with constraints

I've been thinking about a division problem for groups that I haven't found a dynamic programming solution and I'm trying to analyze the complexity of the problem. There is a set of $n$ positive ...
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Problem with proving that $RP \subseteq NP$ : a non-deterministic TM for a language $L \in RP$

I'm having a small issue with wikipedia's proof that $RP \subseteq NP$: An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic ...
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“Given an algorithm, decide whether it runs in polynomial time” is this problem in NP?

This problem is not decidable (reducible to halting problem) but is semi-decidable and therefor verifiable (as those two definitions are equivalent: How to prove semi-decidable = verifiable?). However,...
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How to prove P = NP if problem Π ϵ NP-complete and Problem complement Πc ϵ NP?

How to prove if P = NP if problem Π ϵ NP-complete and Problem complement Πc ϵ NP? OR P = NP if NPC intersects with Co-NPC
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Why is the hitting set problem in NP

I am citing the definition of the Hitting Set Problem from (Gardy & Johnson, 1979): INSTANCE: Collection $C$ of subsets of a set $S$, a positive integer $K$. QUESTION: Does $S$ contains a ...
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Given L1 and L2 in NP, if L1 transforms L2 and L2 transforms HC then L1 NP complete?

Why this Question is False ? NP-complete problems are the hardest problems in NP. if L is in NP-complete then [L must be in NP, all problems in NP can be transformed to L]. Does this mean that L2 ...
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Complete resolution rule for 1-in-k SAT

In CNF SAT, each clause (A or B or C or...) must contain at least one true literal. The resolution rule applies to pair of clauses who have exactly one opposite literal. (A or B or C) and (!A or D ...
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how to prove that $NP \cap co - NP$ = { S | S such that there exist a Strong Deciding Algorithm for S}?

i need to prove that and i find it struggle: given: for deciding problem S: a non deterministic algorithm $A(x)$ is strong deciding algorithm if: $x \in S =>$ fo every run of $A(x)$ returns "Yes"...
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Exact2IS - Question

I have the following question for Exact2IS problem that is defined: $$ \mathrm{Exact2IS} = \{(G,k) \mid \text{$G$ contains exactly two independent sets of size $k$}\}. $$ and I would like to know ...
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Combining 2 problems in NP into one

Say I have a deterministic turing machine which solves decision problem S with oracle access to both problems B, C that are in $NP$. Can S be solved with oracle access to only one problem in $NP$? ...
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Is it possible that Co-NP = P but NP != P

Suppose there exists an algorithm that takes as input an unsatisfiable SAT formula, and never fails to verify it in polynomial time. However, when the input is a satisfiable formula, it doesn't work (...
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Can 3-coloring be reduced to 3-clique?

I'm a slight disagreement with my professor over whether or not a certain reduction is possible. He asked us to reduce the problem of 3-Coloring to the problem of 3-Clique. The problem is that I'm ...

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