Questions about relationships between complexity classes.

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2
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58 views

A particular complexity

Whats is the name for a complexity like $n^{\log \log n}$ ? Is this exactly subexponential, or less than that ?
3
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45 views

Is the minimal number of colors needed to color a graph some fixed number?

Consider to following decision problem: Input: Undirected graph $G=(V,E)$ Question: Is the minimum numbers of colors needed to color the vertices (such that every two adjacent vertices ...
2
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1answer
59 views

Proof of $P^{\text{#}P} = P^{PP}$

I was reading this article on the complexity class $PP$. In the fourth paragraph there is a claim that $P^{\text{#}P} = P^{PP}$ and that it can be proved using binary search. Can anyone please ...
5
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1answer
35 views

Question on NP $\cap$ coNP

I'm struggling with a past paper question and would appreciate any hints: Suppose $L_1, L_2 \in $ NP $ \cap $ coNP and $L_1 \oplus L_2 = \{ x : x $ is in exactly one of $L_1 $ or $ L_2 \} $. Then ...
1
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1answer
44 views

What's wrong here, or, is CNF to DNF conversion in o(exp(n))?

I've been thinking about conversion from CNF to DNF. Assume a "worst case" CNF formula with $k$ disjunctions, each containing exactly $l$ elements and no variable is used twice. Example with $k=3$ and ...
1
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1answer
53 views

Why is NP not trivially equal to Co-NP? (a.k.a. what does Co-NP mean exactly?) [duplicate]

I've been trying to wrap my head around Co-NP, and how it's different to NP, but I am having some trouble. Co-NP is defined by Wikipedia as this: "A decision problem $\mathcal{X}$ is a member of ...
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0answers
39 views

2-depth arithmetic circuits and VP vs VNP

the field of arithmetic circuit complexity is undergoing major discoveries in recent years as mentioned by Fortnow. am looking for a more layman-readable summary: is this new paper Sums of ...
3
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0answers
22 views

A totally-ordered set of functions

When we analyze algorithms using the $O$ notation, we usually use only a small set of the space of all functions. E.g., we use $\Theta(n)$ but not $\Theta(2n)$, as these two are equally well ...
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29 views

Status of $BQP^{NP},NP^{BQP}$

The relation between $BQP$ and $NP$ is an open problem, while it seems that $BQP$ is somewhat lower for $NP$ than the other way round. Is the status of lowness of these problems known?
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1answer
69 views

Are there any natural $\Pi_2^P$-complete problems?

I know that the quantified Boolean formula problem for a formula $$ \psi = \forall x_1 \ldots \forall x_n \exists y_1 \ldots \exists y_n \phi $$ where $\phi$ contains no quantifiers and only the ...
1
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1answer
87 views

PSPACE languages reducible to other PSPACE languages in polynomial space

Intuitively it makes sense that all PSPACE languages are reducible to other PSPACE languages in polynomial space. But how would I go about actually showing this?
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1answer
75 views

What is the relation between NC and P/poly?

I am unable to see a clear explanation of how the classes NC and P/poly intersect or not. (and if they do intersect then how and where? and if not then what is the proof?) I recently was attending ...
2
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0answers
133 views

PSPACE completeness, with different kinds of reductions

PSPACE-complete$_{FP}$ problems are the PSPACE problems such that every other PSPACE problem can be transformed to it with a polynomial time reduction. This class is known as PSPACE-complete. ...
10
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1answer
112 views

Proof of Karp-Lipton theorem

I am trying to understand the proof of the Karp-Lipton theorem as stated in the book "Computational Complexity: A modern approach" (2009). In particular, this book states the following: ...
4
votes
1answer
45 views

What is an $NP^{NP}$-complete problem? [duplicate]

So in this paper I'm reading (https://adamsmith.as/papers/fdg2013_shortcuts.pdf), the authors talk about an $NP^{NP}$-complete problem, in relation to Answer Set Programming. I know what P, NP, etc. ...
1
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1answer
46 views

FNP ⊂ FPSPACE or FNP ⊆ FPSPACE?

It is clear, that NP ⊆ PSPACE holds and that it is unknown if the strict inclusion holds. How is it if one looks at the corresponding functional complexity classes? Does FNP ⊂ FPSPACE hold?
2
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1answer
27 views

Can we separate P and E?

Let $\mathsf E$ be deterministic exponential time with linear exponent. Do we know that the inclusion $\mathsf P\subseteq\mathsf E$ is strict? If so, what's the proof? The time hierarchy ...
7
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1answer
155 views

What do we know about NP ∩ co-NP and its relation to NPI?

A TA dropped by today to inquire some things about NP and co-NP. We arrived at a point where I was stumped, too: what does a Venn diagram of P, NPI, NP, and co-NP look like assuming P ≠ NP (the other ...
3
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1answer
61 views

FP^NP-complete problems

Is there any other standard FP^NP-complete problem other than the Traveling Salesman Problem? For instance, in the canonical propositional logic?
2
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1answer
24 views

Looking for an example of proving space upper bounds for computing functions on a DTM

Like think of the function $f\colon \{ 0,1\}^* \rightarrow \{0,1\}^*$ which maps a binary string string $x$ to say a string of $0$s of length $\vert x \vert ^2$ whre $\vert x \vert$ is the length of ...
0
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0answers
24 views

$k$-query oracle Turing machine (Sipser 9.21)

Question: A $k$-query oracle Turing machine is an oracle Turing machine that is permitted to make at most $k$ queries on each input. A $k$-query oracle Turing machine $M$ with an oracle for $A$ is ...
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0answers
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Proving $CVal$ is $RP$-hard

Let CVal be the language of all $<C,s>$ where $s$ is an $n-$tuple of binary values ($\{0,1\}$), such that $C$ is a variable-free boolean circuit (gates $\wedge$, $\vee$, $\neg$, $0$, $1$), and ...
4
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1answer
173 views

What's the meaning of the name P/poly?

I understand the normal definitions of the P/poly class, but I'm curious how the name came about. The syntax of the name looks like a quotient group, but I can't think of any way to define P/poly ...
2
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1answer
36 views

Definition of $D^P$?

What is the definition of the complexity class $D^P$? In recent papers I sometimes read $D^P$ but could not find a definition of it. Unfortunately Complexity Zoo does not give one, too. Is it the ...
2
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crypto protocols from complexity class

Assume $P=PSPACE$. Then would it be possible to design cryptographic protocols based that is easy to compute from $PSPACE$ but hard to invert from something higher up in hierarchy? A function $f: ...
2
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1answer
105 views

Is it true that $FP^{NP[log\cdot n]} = FP^{NP}$ if $P = NP$?

Is it true that $FP^{NP[log\cdot n]} = FP^{NP}$ if $P = NP$? If I understand the polynomial hierarchy correctly, then, if $P = NP$, all complexity classes collapse to one class. Therefore the above ...
0
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1answer
60 views

Which complexity class $3^{n/3}$

Assuming a problem has complexity $O(3^{n/3})$, Which is its class of complexity ? Despite that it is not as $2^{n}$ ,we can say is an exponential ?
6
votes
2answers
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Is $NP$ “minimal”, i.e. does $\Pi\notin NP$ imply $\Pi$ is $NP$-hard?

Suppose $\Pi$ is a decidable decision problem. Does $\Pi\not \in NP$ imply $\Pi$ is $NP$-Hard? Edit: if we assume there exists $\Pi\in coNP\setminus NP$ then we are done. Can we refute the claim ...
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2answers
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Why do we believe that PSPACE ≠ EXPTIME?

I'm having trouble intuitively understanding why PSPACE is generally believed to be different from EXPTIME. If PSPACE is the set of problems solvable in space polynomial in the input size $f(n)$, ...
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vote
2answers
143 views

How can I use the NP complexity Venn diagram to quickly see which class of NP problem can be poly reducible to another class?

I'm so bad at solving the problem of the type: "If $A$ is an NP-complete problem, $B$ is reducible to $A$, then $B$ is..." That I have to come here and ask these silly questions each and every ...
2
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1answer
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If P = NP, why does P = NP = NP-Complete? [duplicate]

If P = NP, why does P = NP also then equal NP-Complete? I.e. Why would it then be the case that ...
6
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2answers
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Analog of PP for computability rather than complexity?

The complexity class PP can be defined in many ways, one of which involves randomness - a language $L$ is in PP if there is a polynomial-time, randomized TM $M$ such that $w \in L$ if and only if the ...
3
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1answer
67 views

Hierarchy of complexity classes $\bigcup_{c > 0} \mathrm{Time}(2^{c \log^k n})$, w.r.t. $k$

This is a true/false question: For each integer $k > 1$, define the complexity class $\sf QP_k := \bigcup_{c > 0} Time(2^{c \log^k n})$. Then for all integers $k > 1$, $\sf QP_k ...
3
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1answer
163 views

Complexity Classes (P, NP) vs Language Hierarchies (REC, RE)

Is there any relation between the Complexity Classes (like P or NP) and Language hierarchies (like REC or RE) ? Form what I understand: (easy things are the things that can be done in polynomial ...
3
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2answers
241 views

Decision problem which belongs to P reduced to a decision problem which belongs to NP?

Is it possible to have a decision problem $A$ which belongs to P and reduce it to a decision problem $B$ which belongs to NP, i.e. $A \leq_{\mathrm{p}} B$, where $A$ belongs to P, $B$ belongs to NP?
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1answer
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Some questions about NP / coNP / CSP

I need help with the following mock exam questions. True or false? 1.) If a non-trivial $(\neq \emptyset, \Sigma^*)$ finite set is NP-complete, then $P = NP$. True. Every finite set is in $P$ and ...
2
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1answer
101 views

Are there any coP problems

Is there a notion of coP problem? Also is there a notion of every problem being reducible to one problem in P (like 3SAT in NP completeness)?
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1answer
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If P = NP, then is NP = FNP?

I read FP = FNP iff P = NP which makes sense. But if P = NP, does it mean FNP = NP? Intuitively, I think no because P = NP would mean that decision problems in NP would become decision ...
1
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1answer
111 views

Complexity of variation of partition problem

I want to know whats the complexity of the following variant of the partition problem: Partition problem: http://en.wikipedia.org/wiki/Partition_problem Suppose we have one set formed by integers ...
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0answers
28 views

Canadian traveller problem on directed acyclic graphs

What is the complexity of the Canadian traveller problem variant where the only thing that is seen is a single node ahead on a directed acyclic graph so that we cant go back once we go to a new node ...
7
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0answers
58 views

Exponential analogue of NC?

Nick's Class (NC) is the class of problems that can be decided in poly-log time using a polynomial number of processors. I want to know about the exponential analogue, which would cover problems that ...
4
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1answer
63 views

Understanding the Sipser-Gacs-Lautemann theorem

The class $BPP$ contains all the languages decided by a probabilistic Turing machine in polynomial time with probability of success more that 2/3 for every input. The class $\Sigma^p_2$ contains all ...
9
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2answers
215 views

Are there established complexity classes with real numbers?

A student recently asked me to check an NP-hardness proof for them. They performed a reduction along the lines of: I reduce this problem $P'$ that is known to be NP-complete to my problem $P$ ...
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2answers
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Does #$P$-Completeness imply approximation hardness?

Let $\Pi$ be some counting problem which is known to be #$P$-Complete. Does it imply that $\Pi$ is $APX$-hard (i.e. no PTAS for the problem exists unless $P=NP$)?
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1answer
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Is FNP = FEXPTIME if and only if NP = EXPTIME?

It is very well known that if the classes $\sf FP$ and $\sf FNP$ are equal, then also the classes $\sf P$ and $\sf NP$ are equal (see e.g. FNP on Wikipedia). Is it also true that if $\sf ...
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1answer
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exponential lower bound on boolean formula conjunctions, what complexity class? [closed]

This new paper A Lower Bound for Boolean Satisfiability on Turing Machines by Hsieh asserts an exponential lower bound for a TM time complexity on a problem of finding whether a solution exists to a ...
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0answers
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Would $\sf RP = NP$ imply $\sf NP = coNP$?

If $\sf RP = NP$ then the hierarchy collapses to its second level (by the Karp-Lipton theorem). But what about $\sf NP$ and $\sf coNP$? I tried to prove that $\sf BPP$ is contained in $\sf NP$ (the ...
9
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1answer
103 views

What do complexity classes look like, if we use Turing reductions?

For reasoning about things like NP-completeness, we typically use many-one reductions (i.e., Karp reductions). This leads to pictures like this: (under standard conjectures). I'm sure we're all ...
3
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1answer
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Is there a largest class of halting programs?

The halting problem says that a Turing machine cannot decide if another Turing machine halts. However, we know that it is possible to determine if some programs halt. For example, FORTRAN DO ...
2
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1answer
329 views

Are there any PSPACE problems that don't exist in NP-Hard?

The question is in the title, I suppose. I am studying complexity classes, and I understand that NP-Hard is the set of problems that are at least as hard as the hardest problems in NP. Therefore, it ...