Questions tagged [complexity-classes]
Questions about relationships between complexity classes.
344
questions
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Prove that a set is decidable using time constructible function
I'm preparing an exam of theory of computation and I'm very in trouble with some exercise.
Considering a Turing machine $\mu$ of alphabet $A=\{ 0,1 \}$ (we don't know nothing about termination) and a ...
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78 views
Class of languages recognizable by n-bit formulas of size at most $T(n)$
A Boolean (combinatoiral) circuit is a labeled (with the labels: AND, OR, NOT, IN, OUT), directed, acyclic graph, that satisfies:
fan-in=2 for the AND and OR nodes
fan-n=1 for the NOT nodes
fan-...
3
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1answer
63 views
Comparing asymptotic running time of two algorithms $\sqrt n$ and $2^{\sqrt{\log _{2}n}}$
Given two algorithms with their time-complexity $t_a(n)=\sqrt{n}$ and $t_b(n) = 2^{\sqrt{\log _{2}n}}$ and i have to show $t_b(n) = O(t_a(n)) $.
I´ve made a program to check this statement and it ...
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21 views
Proof by padding: $\textsf{TIME}(t_1(n)) = \textsf{TIME}(t_2(n)) \implies \textsf{TIME}(t_1(f(n))) = \textsf{TIME}(t_2(f(n)))$
I've been given the task of proving the statement in the title, which I found out it should be called the translational lemma by means of a padding argument; $f$, $t_1$ and $t_2$ are three ...
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61 views
Proving a pattern exist in a string without revealing where
Some time ago i read the following problem (i don't remember the article from which i read it from) :
"Suppose you are given a picture where the goal is to find waldo (from the game where is waldo), ...
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1answer
31 views
how to proof ${ NPC \bigcap CO-NPC \ne \varnothing then NP = P ? }$
how proof ${\ \ NPC \ \ \bigcap \ \ CO-NPC \ne \varnothing }$ then
${NP = P ? }$
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1answer
65 views
Assume that NP = DTIME(2^sqrt(n)), prove that DTIME(2^sqrt(n)) = DTIME(2^n)
I tried using the padding argument to prove such a thing (as it appeared in Arora's book), but I am not sure how this technique will help me here. I am trying to get to a contradiction to the Time ...
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34 views
Complexity problem reduction?
Let say A and B are two decesion problems where A $\le$ B polinomial reduction is true.
Is this : A̅ $\le$ B̅ also true?
If so, can you show an exemple, if not why?
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2answers
80 views
Does $P/O(1)$ equal to $P$ if solver needs to consider smaller inputs?
Suppose that $F$ is a problem such that for every $n$, there is a program of length $O(1)$, running in polynomial time to $n$, that solves $F$ correctly on all instances of size less than $n$. Can $F$ ...
3
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1answer
100 views
Grover's algorithm on probabilistic classical machines
As a starting point for this question, I came across this question, which AIUI is citing a construction showing how to simulate quantum circuits with a $PP$ algorithm, i.e. implying quantum ...
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30 views
A heuristic for finding an edge cycle cover
I am looking to find a minimum list of cycles in a graph such that their union gives the list of all simple cycles in this graph.
In the example below, here are 4 simple undirected cycles: 1-2-3, 2-3-...
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36 views
Approximate algorithms for class P problems
As a part of my Algorithm course we studied Approximate Algorithms for NP-complete or NP-hard problems, e.g. "set cover", "vertex cover", "load balancing", etc. My professor asked us as an extra ...
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29 views
Cook Levin Theorem (Sipser Proof) (phi move)
In Sipser's proof of the cook levin Theorem the move function (phi move) checks whether a given window is legal. For that we must have an exhaustive set of all possible legal windows to verify that a ...
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35 views
Why is $DSPACE(\log(n)) = NSPACE(\log(n))$ not known?
Here $DSPACE(\log(n))$ is the family of algorithms for which there exists a deterministic Turing machine using $O(\log(n))$ space.
On the other hand $NSPACE(\log(n))$ is the family of algorithms for ...
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1answer
47 views
If a sparse language is NP complete, then are all languages in NP in P/poly?
If a Sparse Language is NP complete, then are all languages in NP in P/poly? I know that sparse languages are in P/poly, but does a polynomial time reduction give an addition to the circuit that is ...
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1answer
73 views
If a sparse language is NP-complete then are all NP languages sparse?
If a Sparse Language is NP-complete then are all NP languages sparse? We say a language is sparse if $\forall n \in \mathbb{N}, |L \cap \{ 0,1 \} ^{n}| \leq p(n)$, for some polynomial $p(n)$?
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1answer
68 views
{0,1}* ∈ P class?
I have the following question about complexity time classes. Given the language $L = \{0,1\}^*$, is it inside the class P or not?
$$ L = \{0,1\}^∗ ∈ P? $$
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1answer
51 views
If an NP complete problem 'A' is polynomial time reducible to another problem 'B' does that imply 'B' is also NP complete?
The following question was asked on a quiz:
Let S be an NP-complete problem, and Q and R be two other problems (that we don't know much about). If we now know that Q is polynomial time reducible (i....
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1answer
66 views
Why log-space reduction is used for NL-completeness while PSPACE reduction isn't used for PSPACE completeness?
NL-Complete languages are defined by Log-space reduction, while PSPACE complete languages are defined by poly-time many-to-one reduction.
According to these posts :
Why not polynomial-space ...
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2answers
165 views
Reducing from NPC to Co-NPC => NP = Co-NP?
In my lecture we learned:
If X is NPC and X in Co-NP => NP = Co-NP
Would it be enough to prove NP = Co-NP if I reduce a ...
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98 views
Is there a polynomial-time algorithm to minimize regular expressions without Kleene closures/stars?
I have read that minimizing regular expressions is, in general, PSPACE-complete. Is it known whether minimizing regular expressions without the Kleene closure (star, asterisk) is in P?
The language ...
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1answer
287 views
Is DISCRETE LOG a NP hard problem?
In cryptography there are two problems which are part of the foundation of modern public key cryptography. Both of them can be solved in polynomial time on quantum computers. I am talking about:
FACT ...
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1answer
58 views
The padding argument in the proof of NTIME(n) ⊆ DTIME(n^1.2) implies Σ2-TIME(n^8) ⊆ NTIME(n^9.6)
In "Computational Complexity, A modern approach", Arora & Barak proof the following claim (Claim 5.11.2):
Suppose that $\mathsf{NTIME}(n) \subseteq \mathsf{DTIME}(n^{1.2})$. Then $\Sigma_2$-$\...
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2answers
135 views
Big-O Notation and Calculus?
I was wondering if there are any calculus relationships implicit in Big-O notation.
For example, an algorithm linear according to Big-O notation reduces the size of the problem by a constant amount ...
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1answer
79 views
Does EXP^EXP = EXP? [duplicate]
Does $\mathrm{EXP}^\mathrm{EXP}=\mathrm{EXP}$?
Here is my thought: $\mathrm{EXP}$ machine can ask $2^{O(n)}$ queries to the oracle, and each oracle would itself solve an exponential time problem in a ...
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0answers
22 views
Which is harder, an NP-complete problem or the Raz-Tal oracle problem?
This is a (hopefully) sharper version of a question that I asked previously.
Which of these algorithms is believed to have a longer asymptotic runtime?
The optimal algorithm guaranteed to solve some ...
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0answers
14 views
Do relativized relations between complexity classes tell us anything about the nonrelativized relation?
The existence of relativized relations between complexity classes seems to often be treated as "circumstantial" evidence about the "true" or "real-world" (i.e. nonrelativized) relation between the ...
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1answer
79 views
Proving that $NPSPACE\subseteq PSPACE$ using the proof of Savitch's Theorem
We were shown a proof of $NPSPACE\subseteq PSPACE$ in class. In short, the proof says:
Let $L\in NPSPACE$.
Then there exists a non-deterministic polynomial space bounded Turing machine $M$ that ...
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1answer
36 views
Complexity of K-Colorful Coloring Problem for a Hypergraph
I searched a lot trying to find a reference for the complexity of K-colorful coloring problem for a hypergraph but I cannot find it. Please if anyone has a reference for the complexity of the problem ...
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$P$ with $SAT[k]$ and $NP[k]$ oracles?
We know $coNP$ is in $P^{NP}$ and so does $coNP$ in $P^{NP[1]}$ and $P^{SAT[1]}$ hold?
Is there a difference between $P^{SAT[k]}$ and $P^{NP[k]}$ at any $k\geq0$?
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45 views
Intersection of decision problems?
Say we have two problems $\Pi_1\in NP$ and $\Pi_2\in coNP$. Where does $\Pi_1\cap\Pi_2$ live?
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1answer
44 views
prove that there is a complete language in $L \cup \{A_{TM}\}$
$A_{TM} = \{\langle M,w\rangle\mid w\in L(M)\}$
$L$ = complexity class containing decision problems that can be solved by a deterministic Turing machine using logarithmic space
Given the language $L ...
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0answers
47 views
Is PSPACE vs NEXPTIME known?
I know that P = PSPACE is a famous open problem, and that EXPTIME = NEXPTIME is also unknown. By the time heirarchy theorem we know that NP is a strict subset of NEXPTIME.
Is anything known about ...
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1answer
26 views
Problem class of assigning N persons to N tasks, zero costs with prefs
I am looking for the general problem class / computational complexity / algorithms for the following problem:
N tasks must be accomplished by N persons. 1 task to be done by exactly 1 person and vice ...
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2answers
20 views
Why do we use worse-case when categorising problems?
Maybe I am wrong, but I read that when we categorise problems in their respective complexity classes, we use worse-case analysis.
Why don't we use the average case?
I imagine we could have a problem ...
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2answers
78 views
DSPACE(f(n)) closed under complement
I think you can create the complementary language that is an element of DSPACE($f(n)$), where $f(n) \geq \log(n)$ by adding a step to the algorithm that reverses the answer. By that the function $f(n)$...
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1answer
127 views
Proving NP completeness of maximal length path
I have this question to answer:
For each node i in an undirected network $G = (N,E)$, let $N(i) = \{j \in N : \{i, j\} \in E\}$ denote the set of neighbors of node $i$ and let $c_e\geq0$ denote the ...
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1answer
33 views
NL-Hardness of Target
When revising for an upcoming exam in complexity theory, I came across this problem on the final part of a question, which I was unable to solve:
$ TARGET = \{<G, t> : t\ is\ reachable\ from\ ...
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1answer
54 views
A simple clarification on polynomial hierarchy
$P^{NP}\subseteq BPP^{NP}$ holds. According to current knowledge $BPP$ is in $\Sigma_2^P\cap\Pi_2^P$ holds. So according to current knowledge is following true?
$P^{\Sigma_2^P\cup\Pi_2^P}\subseteq ...
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0answers
18 views
Why is $BPP^{NP}$ in the polynomial hierarchy?
Why is $BPP^{NP}$ in the polynomial hierarchy?
I know that $BPP$ is contained in $NP^{NP}$, so $BPP$ is inside $PH$. Should I just simply reuse the proof of $BPP\subset NP^{NP}$, feed in each machine ...
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31 views
is this given language in class P? [duplicate]
i was wondering:
is this language in class P?
$NONDISJOINT_{DFA}\:=\:\left\{<A,B> |\:A\:and\:B\:are\:DFAS\:and\:L\left(A\right)\:\cap L\left(b\right)\:\ne \varnothing \right\}$
explanation: a ...
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1answer
32 views
Can CNF with an input string be evaluated in logarithmic space?
I have been trying to solve satisfiability of
{$<c, w>$ | $c$ is a CNF and $w$ is a binary string which satifies the $c$}.
As first looks to me, it is satisiable in linear time ($O(n)$) since ...
2
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1answer
36 views
On the robustness of BQP class
Typically the notion of quantum Turing machine is introduced with its transition function.
$$
\delta:Q\times \Gamma\rightarrow \mathbb{C'}^{Q\times \Gamma\times\{L,R,0\}}
$$
Where $\mathbb{C}'\...
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1answer
25 views
Changing probabilities to 0/1 in definition of class IP
A language $L$ belongs to $\mathbf{IP}$ if there exists $V,P$ such that for all $Q$, $w$,
$$w\in L\Rightarrow Pr[V\leftrightarrow P\text{ accepts }w]\geq2/3$$
$$w\notin L\Rightarrow Pr[V\...
5
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1answer
111 views
If NP is a subset of DTIME[n^O(log n)] then what happens?
If $\mathsf{NP}\subseteq \mathsf{DTIME}[n^{O(\log n)}]$ then what happens? Does it imply $\mathsf{NP}\neq \mathsf{EXP}$? Is there any other consequences such as $\mathsf{BPP}\neq \mathsf{EXP}$? Does ...
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44 views
Reduction of complement from complexity class co-np and p
Let P $ \neq $ NP.
D is in the complexity class co-NP.
B is in the complexity class P.
Let $ \bar{D} $ be the complement of D, then $\bar{D} $ $\leq _ {p} $ B.
Is this statement true or false? My ...
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1answer
40 views
Show: “Checking no solution for system of linear equations with integer variables and coefficients” $\in \mathbf{NP}$
I've been struggling for a while trying to solve this problem:
Show that the following problem is in $\mathbf{NP}$:
Check that a system of linear equations with $m$ integer variables and
integer ...
1
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1answer
69 views
Proof of proposition between Precise Turing Machine and Proper Complexity function
In "Computational Complexity" textbook by C. H. Papadimitriou, p. 141, he proved the following claim.
Proposition 7.1: Let there be a DTM/NDTM M that decides a language L within time/space $f(n)$, ...
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2answers
39 views
NP-hardness does not imply lower bound, strictly speaking?
A problem is NP-hard iff every NP problem can be polynomially-time reduced to it.
Hardness is often intuitively explained as a lower bound. But it isn't, strictly ...
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9 views
$NC$ and $FNC$ oracles low for functional and Stockemeyer classes respectively?
We know $P^{NC}=P$ and $FP^{FNC}=FP$ hold. Do $FP^{NC}=FP$ and $P^{FNC}=P$ hold?
