Questions tagged [complexity-classes]

Questions about relationships between complexity classes.

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70 views

Polynomial time optimization problems belong to which complexity class?

I know that $\mathsf{P}$ class is only defined for decision problems. Therefore, a problem like "Does there exist an $(s,t)$ path of length $k$ in the graph $G$?" is in $\mathsf{P}$. One can ...
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62 views

How to prove that existence of one-way functions implies P≠NP?

Wikipedia: The existence of such one-way functions... would prove that the complexity classes P and NP are not equal. How is this proved?
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Consequence of NP-complete, and DP-complete w.r.t. randomized reductions

If a problem is NP-complete with respect to randomized (polynomial time) reductions, but not with respect to deterministic reductions, then we have P $\neq$ BPP (See Question 2 here and its answer). ...
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1answer
13 views

Different definitions of complexity class DP

I would like to know how different defintions of class DP (also written $D^p$) are equivalent (brief explanations would do). The following are the two definitions I see: Difference between two NP ...
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1answer
23 views

Are problems that are fractions of constraints of NP-complete problems also NP-complete?

We know that Hamiltonian path, clique and independent sets are NP-complete but what about half or the square root of each problem or a fraction of $n$ ? That is, for a graph, $G$ of $n$ nodes, is ...
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77 views

How does strong NP-completeness agree with encoding complexity?

I've recently read about the concepts of weak and strong NP-completeness, but faced a problem in wrapping my head around them. I've understood that problems which have numerical parameters (like ...
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1answer
47 views

What is the relation between $USAT$, $UP$ and $NP=RP$?

Definition: AtmostONESAT: SAT instance having promise of $\leq1$ witness. What is the complexity consequence if an instance of $SAT\in$ AtmostONESAT can be decided whether or not there is a witness in ...
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20 views

Polynomial reduction, #P-hard problems, and approximations

Consider two statements. Statement 1: The problem #3SAT (finding the number of satisfying instances to a 3SAT problem) is #P-hard. Statement 2: Additively approximating #3SAT upto $\pm 2^{n/2}$ error ...
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1answer
39 views

Randomized Version of NP

I came across interactive proofs and randomized computation, in particular, i read about the complexity classes $\text{IP}, \text{BPP}, \text{RP}$, etc. Since the above classes are well-known, I will ...
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1answer
57 views

$coNP$ and $\oplus P$?

Let a non-deterministic machine have at most $2^{t+1}-1$ accepting paths (highest significant bit position is $t$ and lowest significant bit position is $1$). I want to decide if the number of ...
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1answer
34 views

What is the “formula” for “any chipher can be deciphered by a quantum computer”?

There are several quantum complexity classes in different ways analogous to NP: NQP, QMA, and, as I understand, others. P=NP BPP=NP in simple words means "any cipher can be deciphered by a ...
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Does having a similar constraint while reducing a problem to similar problem to prove np hard means they are same?

I have been trying to find the computational complexity of my optimization problem and found that it is Np-Hard. To prove it to Np-Hard, I try reducing it Nurse Scheduling Problem. I am quite confused ...
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1answer
13 views

What is the complexity class of finding vertex cover number of a simple graph?

Suppose we have a simple graph $G$. We know that finding the minimum vertex covering set for $G$ is in the NP-hard class. But, what about the complexity class of finding the size of the set, i.e., the ...
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1answer
10 views

Is it possible to compute the minimum vertex covering set in quasi-polynomial time, by knowing the vertex cover number?

If we know the vertex cover number for a simple graph $G$ denoted by $\tau(G)$, is it possible to find the minimum vertex cover set for $G$‌ in quasi-polynomial time? As I found, we cannot find any ...
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26 views

On classes $\mathsf{\oplus PSPACE}$ and $\mathsf{\oplus L}$?

Savitch's theorem provides $\mathsf{PSPACE}=\mathsf{NPSPACE}$. Is there a class $\mathsf{\oplus PSPACE}$ analogous to $\mathsf{\oplus L}$ and is it known to be equal to $\mathsf{PSPACE}$? If $\...
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33 views

An opposite method of padding argument on N/DTIME complexity class

Is there a method to prove things with longer input in complexity theory? For example, using padding argument it's trivial to show that $\text{NTIME}(n^2) \subseteq \text{DTIME}(n^4) \Rightarrow \...
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46 views

Sufficient condition for a complexity class's closure under NP-reductions?

Let us say that there exists a $\mathsf{NP}$-reduction from a problem $A$ to another problem $B$ when there exists a non-deterministic, polynomial-time Turing machine $T$ such that for each $a \in A$, ...
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1answer
18 views

Is $E^{quasiP}$ equal $E$ or larger?

Let $quasiP$ be the quasipolynomial time complexity class. Is $E^{quasiP}=E$ false? Is $E^{DTIME(2^{(\log n)^k})}=E$ false at every $k>1$?
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35 views

Language in PSPACE that isn't PSPACE-hard and isn't in PH

Can there exist a language L in PSPACE such that L isn't PSPACE-hard and L isn't in the polynomial hierarchy (PH)? Intuitively, it seems like the answer is no, since TQBF is PSPACE-complete, and any ...
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1answer
116 views

What is the definition of Infinitely Often class in complexity

I was reading a paper and I came across the term $L\notin i.o.Dtime(2^{n^c}/n^c)$. What is the meaning of this?
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Naming of Monte Carlo and Las Vegas Complexity classes

For Curiosity Sake, I was wondering if there was some history behind the naming of "Las Vegas" and "Monte Carlo" classes. I've searched on the internet for quite some time now but ...
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1answer
40 views

What is the computational class of a pushdown automaton with real values?

Say there is a push-down automata, in this example I'll use a deadfish-like set: +: increase x by 1 0: set x to 0 ...
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2answers
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Prove that $ln(n)^r \in o(n^p)$ for $p>0$ and $r\in \mathbb{R}$

I am trying to proof $f\in o(g)$ Let be $r,p\in \mathbb{R}$ with $p>0$ We have $f(n)=ln^r (n)$ and $g(n)=n^p$ I have already proofed that $ln(n)\in o(n)$ via l'Hospital $\lim\limits_{n\to \infty}\...
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Complexity class of problem whose running time features binomial coefficient

I've built an algorithm that, starting from an array of $n$ cells and an integer value $s$, builds $\binom{n+s-1}{s}$ vectors (that is, all the ways to add a certain $s$ quantity fully distributed ...
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123 views

What is the smallest time/space complexity class for which no sparse language is hard?

For example, whether there exists $\mathsf{PSPACE}$-hard sparse language an open problem, as it is not yet known whether polynomial hierarchy collapses. But is it a solved problem for larger ...
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1answer
414 views

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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1answer
32 views

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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1answer
28 views

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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1answer
58 views

Is there any consequence to the existence of $\mathsf{PSPACE}$-complete sparse language like with Mahaney's theorem?

Mahaney's theorem states that the existence of $\mathsf{NP}$-complete sparse language would lead to $\mathsf{P = NP}$. Is there any result result regarding the same for the complexity class $\mathsf{...
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3answers
215 views

Why don't passwords prove P != NP?

Pardon my ignorance on the matter but, Verifying passwords = Polynomial (linear) Guessing passwords = Exponential Since each guess has nothing to do with one another, exponential time is best possible ...
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2answers
55 views

What is the smallest time/space complexity class that is known to contain complxity class $\mathsf{SPARSE}$

Is it known if complexity class of all sparse languages is contained within e.g. $\mathsf{EXP}$ or $\mathsf{EXPSPACE}$? Or what is the smallest time or space complexity class that contains complexity ...
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1answer
369 views

Are there any known W[3] or W[3]-hard problems?

We are currently working on a variant of domination parameter and we have shown that it is in W[3] with regard to parameterized complexity. To show it is W[3]-complete, we must show the problem is W[3]...
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19 views

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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1answer
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Conway's Game of Life: Is it really P-complete?

Wikipedia claims that the Game of Life is P-complete (or the decision problem version of it is; the function version, I suppose, would then be FP-complete). Colloquially, P-complete and FP-complete ...
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1answer
101 views

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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76 views

If X is polynomial-time reducible to Y and X is polynomial-time reducible to Z then Y is polynomial-time reducible to Z?

If $X$ is polynomial-time reducible to $Y$ and $X$ is polynomial-time reducible to $Z$, $Y$ is polynomial-time reducible to $Z$? If $X \leq_p Y$ and $X \leq_p Z$ then $Y \leq_p Z$? True, false or we ...
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2answers
93 views

If X is in NP then $\overline{X}$ is in NP. True, false or “we don't know”? Why?

If X is in NP then $\overline{X}$ is in NP. True, false or "we don't know"? Why?
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1answer
120 views

If X is polynomial reduction to Y and Y is in NP, then X is in NP? [duplicate]

If X is polynomial reduction to Y and Y is in NP, then X is in NP? Is this true, false or "we don't know"? Why?
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48 views

Complexity of approximating a function value using queries

I am looking for information on problems of the following kind. There is a function $f: [0,1] \to \mathbb{R}$ that is continuous and monotonically-increasing, with $f(0)<0$ and $f(1)>0$. You ...
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1answer
62 views

Complexity of generating power sets

Suppose I have two sets $A$ and $B$ containing integers. Let $B'$ be the power set of $B$. Then suppose I have an algorithm that enumerates all possible pairings of elements in $A$ and $B'$ to apply a ...
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1answer
43 views

A question regarding definition of Deterministic Subexponential Time (SUBEXP)

First Look at the definition of SUBEXP from Complexity Zoo: SUBEXP: (Deterministic Subexponential-Time) The intersection of DTIME($2^{n^\epsilon}$) over all $\epsilon$>0. (Note that the algorithm ...
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1answer
74 views

Arthur-Merlin protocol

I recently learned about the Arthur-Merlin protocol, and we defined the complexity classes $AM,MA$. We have also seen that there exists a theorem stating that $AMAMAM...AM=AM$, however we have not ...
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1answer
74 views

Is finding solution to a system of 2SAT equations seperated by OR (DNF form) in NP

I want to know if finding solution to a specific number of 2SAT equations sepearted by OR gate (DNF form as below) is in P or NP. The equation has total n variables and each clause is a 2SAT equation ...
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35 views

If $PSPACE^{SAT}=PSPACE$ and $PSPACE \subseteq EXP$, then why does $EXP^{SAT}$ not necessarily equal to $EXP$?

I read the following claim: $PSPACE^{SAT}=PSPACE$ $EXP^{SAT}$ is not necessarily the same as $EXP$ The first claim makes sense; $PSPACE \subseteq PSPACE^{SAT}$ trivially, and for any language $B \in ...
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1answer
41 views

why is $\Pi_2$ smaller than $NP\cap coNP$

Consider the language $A=\{(\phi_1, \phi_2) | \phi_1 \in SAT, \phi_2\in \overline{SAT} \}$. What is the smallest class that $A$ is known to belong to? Apparently, the answer is $\Pi_2$, although I ...
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25 views

If $NTime(2^n) \subseteq DTime(n^n) $, then what can you conclude about $DSpace(n^n)$?

Assume $NTime(2^n)\subseteq DTime(n^n)$, what can you conclude about $DSpace(n^n)$? I don't know if this is the correct approach, but here was my attempt at an answer: Let $A \in DSpace(n^n) $ and ...
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1answer
34 views

Complexity of class finding selection of entries in matrix

Suppose I have a matrix with entries either $x$ or $y$, where the number of rows = number of columns = $n$. If I want to select/circle $n$ entries such that for each row, only exactly one is circled, ...
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Can you give an example of a problem in $EXP^{RE}$ but not In $P^{RE}$

Can you give an example of a problem in $EXP^{RE}$ but not In $P^{RE}$?
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1answer
104 views

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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18 views

Are there any “natural problems” which are known to be NPI under weak assumptions

Are there any "natural problems" which are known to be NPI under weak assumptions. By weak assumptions I mean something like $P \neq NP$ or $NP \neq Co-NP$

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