Questions related to the (computational) complexity of solving problems

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Shaefer's Dichotomy Theorem

Could you please resolve a confusion with Schaefer's theorem for me? Namely, why does it not imply many problems in P are NP-complete? For example, primality testing surely cannot be reduced to one of ...
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42 views

“Balancing” positive and negative literals in 2-sat

I saw in an answer to this post that it is possible to construct 3-sat clauses with extra variables such that the number of positive and negative literals for each variable are equal. Does anyone ...
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20 views

Complexity of cubic graph decomposition

I am aware that deciding the existence of decomposition of a cubic graph into edge disjoint claws is polynomial time solvable. What is the complexity of deciding the existence of decomposition of ...
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Existence of randomized reduction but no deterministic reduction

What is the consequence to complexity theory of having a randomized reduction from an NP-complete problem to problem $\Pi$ while there is no deterministic reduction from an NP-complete problem to ...
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1answer
75 views

Complexity of a SAT related problem

Given a set of (propositional) formulae $\Phi$, two formulae $\phi$ and $\xi$, determine whether there exists $\Psi\subseteq \Phi$ such that $\Psi\models \phi$ and $\Psi\not\models \xi$. Question: ...
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1answer
24 views

P/Poly class - undecidable lanauge

I didn't understand some things about $ P/POLY$ class, and I will be thankful if you could help me. as I learned in class, a turing machine M accepts language L with advice $ {a_n} $ if: M(x,$ a_|x| ...
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1answer
37 views

Reduction of SUBSET-SUM to SET-PARTITION [duplicate]

There is a similar question that has been asked, but my question addresses particular detail of an answer. I am trying to reduce SUBSET-SUM to SET-PARTITION. I found the following description: ...
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34 views

Question as regards a proof of the Time Hierarchy Theorem

I'm referring to the proof outlined here (and wikipedia.org): https://proofwiki.org/wiki/Deterministic_Time_Hierarchy_Theorem In my understanding, if I relaxed the conditions such that $K$ decides ...
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27 views

Complexity of finding these original parameters

I am given (or rather, generate randomly) three positive integers $a, b, c$. I want to know if there exist integers $m \ge 2, s \ge 1$ such that $ms+m = a, ms+1 = b, 2s+1 = c$. If there are multiple ...
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67 views

List Sort: cannot determine complexity for a new sorting algorithm [closed]

I need some help in determining complexity of an algorithm me and my friend had written a couple of years ago. Its a new way to sort data in a fast manner using lists. To give you an idea of how fast ...
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1answer
55 views

Fast algorithm for clustering groups of elements given their size/time

I don't know if there is a canonical problem reducing my practical problem, so I will just try to describe it the best that I can. I would like to cluster files into the specified number of groups, ...
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1answer
62 views

Computational complexity based in the input size

In an answer to a previous question Luke Mathieson wrote: A place where this might fall down is when you are working with numbers. As a number with magnitude $m$ can be encoded in $n=O(\log m)$ ...
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28 views

Question about the Simon's algorithm

This comes from trying to understand the "Simon's algorithm". So we have a set of $2^n$ kets $|x_i \rangle$ one each for $i \in \{0,1\}^n$. Each $x_j \in \{0,1\}^n$. And we have the further ...
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1answer
17 views

How many inputs does the Hadamard gate have?

Look at the diagram in the middle of page 6-3 here, http://stellar.mit.edu/S/course/6/fa14/6.845/courseMaterial/topics/topic3/lectureNotes/qctlec6/qctlec6.pdf I am confused as to how should one think ...
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1answer
49 views

How to Prove NP-Completeness of Minimum Crossing Problem?

In graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. (from wikipedia) I know that the problem of counting the ...
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1answer
46 views

Hardness of problem related to number of subsets that satisfy a particular property

I have the following algorithmic problem. I am given a set of elements. Each element has a set of properties. For example: ...
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1answer
27 views

About the definition of “differential privacy” in communication complexity

In the context of communication complexity I see a definition of differential privacy which isn't totally clear to me as to why it makes sense. So the two parties $A$ and $B$ draw two strings $X$ ...
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Is L closed under linear-time reductions?

L is as usual the complexity class DSPACE($\log n$), of languages decidable using a deterministic Turing machine using logarithmic workspace. Is L closed under linear-time reductions? It is ...
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73 views

Are there any non-naive parallel sparse matrix multiplication algorithms?

I was wondering about a problem in analyzing a social network (counting friends-in-common between all pairs of members) that requires squaring its adjacency matrix, and started reading up on ...
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13 views

Renyi entropy and differential privacy

In an $(\epsilon,0)$ differential privacy mechanism what does the $\alpha-$Renyi entropy of the output measure? What does it signify about the mechanism?
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41 views

How is a communication protocol a mechanism?

Given a finite set $\Sigma$ and a positive integer $n$, a mechanism is a set $\{ \mu_x \vert x \in \Sigma^n \}$ such that $\mu_x$ is a probability measure on some $\sigma-$algebra for each $x$. Now ...
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18 views

Counting words: performance of loop vs. list comprehensions [duplicate]

I am no computer scientist, so I can be naively missing some commonly implemented optimization strategies but... Looking at two ways of counting words in a text (shown below), I tend to think that ...
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3answers
226 views

Why is the addition function exponential for k-bit integers providing only zero, equality and the successor functions?

I'm currently reading the elements of programming book and have come across a section I don't quite understand A computational basis for a type is a finite set of procedures that enable the ...
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Reduction NP-Complete with graph undirected [duplicate]

Given a graph undirected $G=(V,E)$ a subset $I$ of $V$ is indipendent for each couples of vertices u,v in $I$ and {$u,v$} is not in $E$. Prove that the language $L$={$<G,k>$: $k$ is a positive ...
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1answer
27 views

How can I efficiently find the optimal order to apply special offers to a shopping cart?

Given a list of items which represent items in a shopping cart, and a list of available special offers which replace one or more regular items to lower the cost of those items, how can I decide the ...
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39 views

Which of the two properties isn't satisfied?

Show that the following sequence of function $\Phi_n$ is not a measure of complexity: $\Phi_n(x)=\left\{\begin{matrix} \text{ nr of commands } m \text{ that TM } T_n \text{ executes with ...
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Cobham's characterization of FP

Does anyone know of an accessible introduction to Cobham's model independent characterization of FP and it's equivalence to the standard definition using Turing machines? The best source I could find ...
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32 views

Complexity calculation using a recurrence relation [duplicate]

I just can't solve this problem, I'm new to reccurences. I have this recurrence $T(n)=n*T(n-1)$ $T(1)=1$ The second term will be: $T(n-1)=(n-1)*T(n-2)$ And so on. It's complexity is O(n!) but i ...
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1answer
258 views

If one-way functions exist are we definitely using them?

I know that if one-way functions exist then there are certain universal one-way functions that exist, but to my knowledge they are too impractical to implement (which is the main reason why they are ...
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39 views

Constructing languages in NPI other than through Ladner's Theorem

I have seen proofs of Ladner's theorem which detail the construction of languages in NPI assuming P $\neq$ NP. However, I was wondering if there are any other constructions using the fact that sparse ...
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59 views

Suppose P = NC - what then? [duplicate]

Suppose tomorrow someone discovered a proof that P = NC. What would the consequences for computer science research and practical applications be in this case?
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P vs NP and the Time Hierarchy

Assuming P $\neq$ NP, is it possible that there exists a $k$ such that for all $j$, $\textsf{DTIME}(t^j) \subseteq \textsf{NTIME}(t^k)$? There reason I ask is that I assume P = NP implies that for ...
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29 views

What is the relation between differential-privacy mechanism and entropy?

Why do differential-privacy people care whether or not the noise function saturates the lower bound of Shannon entropy? For example : Laplace distribution that is used to model the noise function ...
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2answers
61 views

Complexity-theoretic difficult of checking the value of $\pi(x)$?

The prime-counting function, demoted $\pi(x)$, is defined as the number of prime numbers less than or equal to $x$. We can define a decision problem from $\pi(x)$ as follows: Given two numbers ...
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1k views

Why do Shaefer's and Mahaney's Theorems not imply P = NP?

I'm sure someone has thought about this before or immediately dismissed it, but why does Schaefer's dichotomy theory along with Mahaney's theorem on sparse sets not imply P = NP ? Here's my ...
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45 views

Do problems in P only reduce to NP and coNP problems?

Consider the languages $B,C,D$, such that $B\le_p C$ and $B\le_p D$. Statement: $B\in P, D\in NP, C\in coNP$. Is the statement true for every $B,C,D$? I know that the answer is no and I have ...
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NP-Hard vs NP-Complete Why NP-complete so important? [duplicate]

A problem $L$ is NP-complete when:- $L\in \text{NP}$ For every problem $L' \in \text{NP}$, $L'$ is polynomial time reducible to $L$ When at least property 2 is satisfied for a problem $L$ (but ...
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Why is it true that $NP \ne coNP \implies X = \emptyset$?

Let the class of languages $$X = \{ L \ | \ L\in NPC \land L\in coNPC\}$$ Why is it true that $NP \ne coNP \implies X = \emptyset$?
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$A$ is finite, $B$ is NPC - When there's a polynomial reduction from $A$ to $B$?

$A$ is finite, $B$ is NPC - When there's a polynomial reduction from $A$ to $B$? Basically, I've understood that if $A$ is finite, then there's a reduction for every $B$ which isn't trivial ...
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Calculating Time Complexity of Quadratic Diophantine Equation

The particular quadratic Diophantine equation: $$ R(a,b,c) \Leftrightarrow \exists X \exists Y :aX^2 + bY - c = 0 $$ is NP-complete. (a, b, and c are given in their binary representations. a, b, c, ...
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2answers
70 views

What is practical difference between NP and PSPACE-complete?

Here's something that has puzzled me lately, and perhaps someone can explain what I'm missing. Problems in NP are those that can be solved on a NDTM in polynomial time. Now assuming P$\,\neq\,$NP, ...
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51 views

Prove that $S_2$ is closed under union and complement

I'm having trouble proving that $S_2$ is closed under union and complement, even though in this Wikipedia article it says that: It is immediate from the definition that $S_2$ is closed under union ...
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1answer
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Solving the graph colourability problem in polynomial time if the equivallent decision problem is in $P$ [duplicate]

For the graph colourability problem, we are given a graph and our goal is to find a colouring of the graph with the fewest possible number of colours so that no two adjacent vertices have the same ...
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Why are decision problems easier than the equivallent optimization problems?

Suppose that we have an optimization problem defined as follows: $OPT$ = Given an input string defining a set of feasible solutions $F$ and an objective function $f$, find $x\in F$ maximizing $f(x)$ ...
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if P=NP then $L\leq L'$ for all languages [duplicate]

How can I prove that if P=NP then for each non-trivial language $L,L'\in NP$ there exists a polynomial reduction $L\leq L'$?
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Multiple FPT Parameters

The class $FPT$ (fixed-parameter tractable) is defined here. However, there is only one "parameter" that is studied from the given problem/language. Is there an equivalently defined class that can ...
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139 views

Could an NP-hard problem have a mechanical or physical solution method?

Is there any NP-hard problem that we can find a mechanical "polynomial time" solution to? For example, suppose we construct a graph out of something physical, e.g. we have have pipes through which we ...
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2answers
109 views

How can P=NP relate to creativity and proof automation, as said by Scott Aaronson?

I read several times of Scott Aaronson saying that P=NP implies that human creativity is boring and something like that, and that P=NP has something to do with proof automation. I don't get his ...
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1answer
25 views

(operationalizable) Cost measure for small problems

For what I know of complexity measures in CS, they are aimed at rather large problems. With today's computing power, most people don't care about comparing the complexity of simple problems as they ...
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1answer
45 views

Computational complexity of function $U^V$ [duplicate]

Given $(U,V)$ two integers of finite size, I have a question about the complexity of calculating the $V^U$, i.e. $V$ raised to the power $U$. Is their a polynomial time algorithm to do this? If not is ...