Questions tagged [np]

NP stands for Nondeterministic Polynomial time.

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Is the knapsack variant with small profit and unlimited repetition of items NP-hard?

Consider the unbounded knapsack problem where we are given $n$ items of integral weights $w_i$ and integral profits $p_i$ . Given a max weight $W$, the goal is to maximize $\sum_i x_ip_i$ subject to $\...
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Implications of proving NP=RP on complexity theory

Edit: As indicated below by Mahdi Cheraghchi and in the comments, the paper has been withdrawn. Thanks for the multiple excellent answers on the implications of this claim. I, and hopefully others, ...
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239 views

When was co-NP introduced for the first time?

My best finding is Pratt's 1975 article. Is there any earlier mention of co-NP?
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Hamiltonian cycle vs co-NP [closed]

I am trying to understand co-NP and its implications properly. The French Wikipedia page describing co-NP provides the "complementary" version of the Hamiltonian cycle in co-NP as follows: <...
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Who introduced the notion of Non-Deterministic Polynomial time?

In his 1972 paper, Richard Karp provides a definition of NP (section 3, definition 4, p.91). It this the original definition of the class NP or is there a previous one? Edit Edmonds mentions the idea ...
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I believe I am able to prove that $NP\subseteq P$ any idea how I show that $P\subseteq NP$ [closed]

I believe I am able to prove that $NP\subseteq P$ any idea how I show that $P\subseteq NP$? I believe it will revolutionize our modern perception about computability.
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Polynomial-time reducibility of Primality and 3-SAT

Is 3-SAT $\leq_{p}$ Primality? And/or is Primality $\leq_{p}$ 3-SAT? I think the answer is no and yes, respectively, but I'm not sure. Any help would be appreciated. Thank you.
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is this selection problem np-hard? [closed]

Give $n$ clusters $C=\{C_i\}_{i=1}^n$ where each cluster consists of a set of similar points, i.e., $C_i=\{c_j\}_{j=1}^{|C_i|}$. The similarty between two points $c_i$ and $c_j$ is denoted as $w(c_i,...
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List of NP-Complete graph problems/ properties?

Is there a good source to find various decision problems on graph and networks? For a project I'm doing it'd be useful to be able to look at lots of different problems. Is there a good source for ...
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Understanding non-equivalence of proof lengths according to proof systems

Here, in section 4.3, Fortnow says: But to prove P != NP we would need to show that tautologies cannot have short proofs in an arbitrary proof system. I am ...
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Fagin's Theorem implications

I was going through Fagin's Theorem (summary on wiki) and if I understood correctly Existential Second Order Logic (ESO) can be used to represent any NP problem, the same can be said for a Non-...
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Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
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Are there any known languages in the intersection of NP and co-NP but not in P? [closed]

We currently don't know the relationship between NP and co-NP, but would it be possible to show whether the intersection is equal to P? I can't think of any languages in both NP and co-NP, but not in ...
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Complexity of a problem over acyclic context-free grammars

Let $G$ be an acyclic, context-free grammar over a fixed alphabet $\Sigma=\{a_1,\dots,a_k\}$ with the restriction (without loss of generality) that $|w|=2$ for each rule $A\to w$ in the grammar. ...
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Have people looked for parameterized algorithms for problems that are not in NP?

Are there problems that are not in NP (e.g., NEXP-complete problems) but admit FPT algorithms for a reasonable parameterization (and specifically, the standard parameterization of a problem -- the ...
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“Berman-Hartmanis Conjecture Separates NP From All Super-Poly. DTIME Classes” — Worthy of arXiv.org?

Do you believe this paper is worthy of arXiv.org? I have searched via Google, and to my knowledge, no one else has this result. I'm not asking you to fully scrutinize the paper, I'm just asking if you ...
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Algorithm whose running time depends on P vs. NP

Is there a known, explicit example of an algorithm with the property such that if $P\neq NP$ then this algorithm doesn't run in polynomial time and if $P=NP$ then it does run in polynomial time?
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Is satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ NP Complete?

Question I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete. Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$. Where $\...
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Are these problems in NP class?

${\bf New\ version}$ [Version 1.2] Let $f: \mathbb{N} \to \{0,1\}$ be a computable function, ${\bf Fin}(\mathbb{Z})$ be the set of all finite subsets of $\mathbb{Z}$, and $W: {\bf Fin}(\mathbb{Z}) \...
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1answer
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NP completeness of classes of spanning trees

I am teaching a complexity course, and I want to give some examples of similar looking problems such that one is in P, and the other is NP complete. This made me think of the following problem: does ...
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Path in a graph with durations [closed]

I have the following problem: given a directed graph $G=(V,E,d)$, where $d:V\to\mathcal{I}(\mathbb{Q}_0^+\cup\{+\infty\})$ (here $\mathbb{Q}_0^+$ denotes the set of non-negative rationals and $\...
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Complexity of a variant of partition problem

Motivated by this post, Strongly NP-complete variants of subset sum or partition problem, I am interested in this variant of partition: Given a solution to balanced partition problem (both parts have ...
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Is $NEXP^{NP}$ known to not be contained in $NP/poly$?

To the best of my knowledge, it is known that $NEXP^{NP} \nsubseteq P/poly$, but it's still not known if $NEXP \nsubseteq P/poly$. For more info, see "Superpolynomial circuits, almost sparse oracles ...
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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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Reduction of graph chromatic number to hypergraph 2-colorability

I'm following this paper titled "Coverings and colorings of hypergraphs" by Lovasz 1973, which is referenced in Garey and Johnson's Computers and Intractability, for the Set Splitting Problem. In this ...
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What are some known methods for showing that a class has no complete problems?

The only way that I know of is the way that you can show that $RE \cap coRE$ does not via diagonalization. Mostly curious because if $NP \cap coNP$ has no complete problems then $P \neq NP$. I tried ...
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Co-Partition Problem: Why is this proof for it being in NP wrong? [closed]

So I have this wrong proof that the problem "Co-Partition" is in NP. I know the proof is wrong because I've encountered it in an educational environment and was told that it's not working. I don't, ...
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Why does not the definition of NP problems care about the complexity of guessing? [closed]

I have a question regarding the definition of NP problems. According to that, a problem is in NP if one can guess a certificate of polynomial size in polynomial time. However, this definition does not ...
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Is $\sf{P^{NP \cap coNP}} = \sf{NP \cap coNP}$?

If it is unknown, are there reasons to believe that they might not be equal?
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Are There Highly Symmetric NP- or P-complete Languages?

Does there exist $L$, an NP- or P-complete language which has some family of symmetry groups $G_n$ (or groupoid, but then the algorithmic questions become more open) acting (in polynomial time) on ...
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Euclidean TSP in NP and square root complexity

In this lecture notes by Ola Svensson: http://theory.epfl.ch/osven/courses/Approx13/Notes/lecture4-5.pdf, it is said that we don't know if Euclidean TSP is in NP: The reason being that we do not ...
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(How) Could we discover/analyze NP problems in the absence of the Turing model of computation?

From a purely abstract math/computational reasoning point of view, (how) could one even discover or reason about problems like 3-SAT, Subset Sum, Traveling Salesman etc.,? Would we be even able to ...
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Is there an NP-complete language that contains precisely half of the n-bit instances?

Is there a (preferably natural) NP-complete language $L\subseteq \{0,1\}^*$, such that for every $n\geq 1$ $$|L\cap \{0,1\}^n|=2^{n-1}$$ holds? In other words, $L$ contains precisely half of all $n$-...
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Is there any relationship of hardness between the two problems?

Assuming F(x,y,D) is a function, and we can evaluate it in polynomial time with input x, y and D. Consider the problem P1: With D as input, computes $(x^*,y^*)=argmax_{(x,y)}F(x,y|D)$ where x and y ...
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Questions regarding SETH

I read about the strong exponential time hypothesis, which states (as far as I understand) that SAT problem cannot be solved in running time $O(2^{\epsilon n})$ for any $\epsilon < 1$, where $n$ is ...
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Is sparse embedding of a NP-complete problem in a polynomial problem NP-complete?

Consider the following problem P: Input is a finite graph G. If the number of vertices in G is 2^2^i for some integer i, then output a minimum vertex cover of G; otherwise output empty set. Can I say ...
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About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
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Does p-isomorphism preserve phase transition?

Consider two NP-complete languages that are polynomial-time isomorphic. If we know that one of them exhibits phase transition (with respect to some order parameter), does this imply that the other ...
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Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis

The Berman Hartmanis conjecture which formally states that there is an isomorphism for two $NP$ complete languages $L_{1}$, and $L_{2}$, the isomorphism is a bijective function $f()$ such that $f()$ ...
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An NP-complete variant of factoring and relation to factoring [closed]

After reading this post An NP-complete variant of factoring. I come up with a question. To summerize the post, we have the factoring problem (F) which ask for a number $p$ that is prime and divides ...
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Intersection of languages in NP

Can intersection of two languages in NP which are not NP complete be NP complete? Can intersection of two languages in coNP which are not coNP complete be coNP complete? Can intersection of two ...
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Graph theoretic restriction to Proofs in Proof Complexity Theory

Proof complexity is a most basic area of computational complexity theory. An ultimate purpose of this area is to prove $NP\neq coNP$, that is, any prover cannot give a proof of unsatisfiability of ...
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non deterministic algorithm always find the right solution? [closed]

i'm confused since i see some books mention the NP time algorithms as "very lucky" that means it always finds the right path? also can someone explain the coming points to me please... A ...
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Are ill-posed inverse problems in NP?

I'm a physicist who works on inverse problems; I'll explain what these are by means of an example. Consider an object whose refractive index is known; then, the problem of computing scattered ...
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Are there any known NP problems which are conjectured to be exponentially hard on average?

ETH states that SAT cannot be solved in the worst case in subexponential time. What about average case? Are there natural problems in NP that are conjectured to be exponentially hard in the average ...
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Natural NP-complete problems with high density?

(This question is related to a previous one, see the discussion in "Almost easy" NP-complete problems, but it may also be of independent interest, so I post it as a separate question.) Let ...
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Relationship between context-free/decidable languages and NP [closed]

As far as I understand all languages in NP are decidable. But not all decidable Languages are in NP, because NP only contains decision problems. Are there also decision problems that are decidable ...
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Is there a non-deterministic linear time algorithm for CNF-SAT?

The decision problem CNF-SAT can be described as follows: Input: A boolean formula $\phi$ in conjunctive normal form. Question: Does there exist a variable assignment that satisfies $\phi$? I'm ...
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What do dichotomy theorems feed on?

It is well known that certain classes of NP-problems have dichotomy theorems, which guarantee that every task in the class is either NP-complete or is in P. The best known such result is Schaefer's ...
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Functional oracles

In the traditional oracle Turing machine, the oracle is specified as a decision problem. Roughly speaking, one puts a string in the oracle tape, and asks whether it is true or false. I am wondering ...