Questions tagged [complexity-theory]

Questions related to the (computational) complexity of solving problems

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Show that NL ⊆ P

Can somebody please explain this in simple terms. I don't understand Sniper's way to explain it. He says this is a corollary to PATH is NL-complete which I do not understand either. Can somebody ...
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22 views

Finding a kernel for 1-Bounded degree deletion of $O(k^2)$ vertices [duplicate]

In 1-BOUNDED DEGREE DELETION problem, we are given an undirected graph G and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the maximum vertex degree ...
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170 views

Why does a polytime hitting set generator derandomize RP?

I am reading Goldreich, Vadhan, Wigderson: Simplified Derandomization of BPP Using a Hitting Set Generator and trying to understand the result that polytime hitting set generators (HSGs) would not ...
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50 views

Is the class $EP^{FewP}$ in $NP$ and $P^{UP}$?

https://complexityzoo.net/Complexity_Zoo:E#ep defines the class $EP$ as the set of problems solvable by an $NP$ machine such that if the answer is $NO$ all paths reject and the answer is $YES$ exactly ...
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Is this solution for the Partition Problem right?

I've tried to solve the Partition Problem for 2 subsets minimizing the difference in the sum between the two subsets. I've coded the following: ...
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1answer
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Showing that a problem in $NP$ admits two distinct verifiers that satisfy additional constraints

Show that any $S \in NP$ has 2 different polynomial-time verifiers $V_1, V_2$ such that, for all $x,y$, the following conditions hold: If $V_1(x,y)=1$ then $V_2(x,y)=0$ If $V_2(x,y)=1$ then $V_1(x,y)=...
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1answer
23 views

Asymptotic notation for summations

I am struggling to understand why this property of asymptotic notation is true
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1answer
13 views

Padding a 2SAT clause

In http://web.mit.edu/neboat/www/6.046-fa09/rec8.pdf, I see that they pad a 2SAT clause $(x\vee y)$ to make it a 3SAT clause by writing $(x\vee y\vee p) \wedge (x\vee y\vee \neg p)$. Why doesn't $(x\...
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70 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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25 views

Reduction from Independent Set with fixed vertex to Independent Set

I was looking to solve this reduction, but I dont see how to construct the new graph. It seems very simple but I'm not capable of do it. I give you the complete explanation about this reduction. We ...
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1answer
181 views

NOT satisfiable 3SAT instance certificate

Given a NOT satisfiable 3SAT instance, that we say $S$. Suppose that $M$ is a minimal subset of clauses of $S$ such that $M$ is NOT satisfiable. Say $X$ the subset of variables of $S$ that belong to ...
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1answer
31 views

Proof that a relation is in FP

How we can prove that the relation: $R= \left\{0,1\right\}^*\times \left\{0,1\right\}^* \in FP$ I understand that we need to find a polytime algorithm to decide whether $(x,y) \in R$ since $(x,y)\in R=...
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Is the multiplicative constant in the Big O notation are ignored because of Linear Speed-Up theorem?

I just want to know if Big O notation was used as a consequences of the Linear speed up theorem or not. For me I guess the answer is yes. For example, if we didn't have a linear speed-up theorem, then ...
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NP as the set of polynomial verifier and set of decidable problem in nondeterministic polynomial time

My initial idea is to show that the equality of the sets by first proving (a) ==> P is subset of NP and secondly, (b) <== NP is subset of P?
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Iterated multiplication of permutation matrices

Given $m$ matrices of size $n\times n$ each of which is promised to be a permutation is it in $\mathit{quasiAC}^0$ or $\mathit{AC}^0$ to multiply the permutations where $m=\mathit{poly}(n)$ $m=\...
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Prove lower bound on boolean circuit

Given matrix $A \in \{0,1\}^{n \times m}$ with $n$ rows and $m = 2^n - 1$ columns. Where $j$-th column is binary decomposition of $j$ ($j = 1 \dots 2^n - 1$). For example, if $n = 3$: $ A = \begin{...
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33 views

What computational complexity classification does autonomous/self-driving cars belong to?

Self-driving car technology continues to attract popular attention and interest in today's media, but how would a computer scientist explain the theoretical nature of the problem? For example, we are ...
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Worst-Case Complexity of Quantifiers in Thompson's Construction

My understanding is that an NFA compiled using Thompson's Construction should have a running time that is linear in the length of the input string, with a space complexity that is linear with the ...
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1answer
118 views

Representing FP relations using witnesses

Let a relation $R$ be A-polynomial if there is a polynomial $p$ and a polytime algorithm $A$ such that for each $x$: If $(x,y) \in R$ then there is a witness $w$ of length at most $p(|x|)$ such that $...
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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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1answer
49 views

Any problem in P can be reduced to the language of odd integers

Given $A=\left\{n\in \mathbb{N} \mid \text{$n$ is odd}\right\}$, we want to prove that if $S \in P$ then there is a Karp reduction from $S$ to $A$. My attempt: If $S \in P$ we can solve $S$ with a ...
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What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?

In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
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44 views

Prove/Disprove: NP is closed under “mixed” complexity

Let $\displaystyle S_{1} ,S_{2} \subseteq \{0,1\}^{*}$, we say $\displaystyle x\in S_{1}°S_{2}$ if it's of the form $\displaystyle x=x_{1} x_{2} ...x_{n}$, for $\displaystyle n$ even, such that: $\...
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Are there more succint algorithm for translating NFA's to DFA's?

When translating an NFA to its deterministic equivalent, we get an exponential blowup due to the powerset construction method. I tried to search but couldn't find an appropriate question regarding ...
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83 views

Is the base of natural logarithms $e$ efficiently computable?

Is $e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} +\ldots$ efficiently computable? To be more specific, we say that a real number $r$ is efficiently computable if there is a ...
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Scott Aaronson's Proof of $\textbf{BPP} \subset \textbf{P/poly}$

The proof is in the image below, taken from "Quantum Computing Since Democritus": Here's what I don't totally get: my understanding of random algorithms is that randomization is not done ...
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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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50 views

A data structure that supports efficient insertion of arbitrary integers, value lookup, and removal of all integers smaller than a given threshold?

I'm looking for a data structure that supports efficient: Insertion of arbitrary integers. Value lookup (given a particular value needle, return true if it's ...
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37 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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55 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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What will be a space complexity of a conversion of a sorted array to a balanced BST in case of a recursive solution?

What will be a space complexity of a conversion of a sorted array to a balanced BST in case of a recursive solution? Could you, please, check my reasoning. Is it correct? Here is my algorithm for ...
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18 views

What is uniformity in Boolean circuits exactly?

I have two questions on Kaveh's answer to Definition of uniform boolean circuit : Kaveh mentions that the input is in unary encoding. In the definition it says the input is $1^n$, afaik $1^n$ is a ...
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36 views

is advice function and an oracle the same thing?

In the context of P/poly complexity class, an advice function is mentioned. How is the advice function different than an oracle(/certificate)? https://en.wikipedia.org/wiki/Advice_(complexity) https://...
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O(n+m) vs O(n+2m) time complexity

Are $O(n+m)$ and $O(n+2m)$ the same? If $m>n$, then both complexities are $O(m)$. Likewise, if $n>m$, then they are $O(n)$. Is this correct?
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Two-color hitting set

Let $n,m$ be a positive integer with $m\ge 2n$. In the set $M=\{1,2,\dots,m\}$, some elements are black and others are white. Given $2n$ nonempty subsets of $M$, the task is to determine whether there ...
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23 views

What is the generated grammar for this language?

I want to construct a regular grammar that generates words that contain both "ab" and "bc" as subwords with the alphabet of the terminal symbols {a,b,c} My solution so far is G=(Vn=...
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Give the Regular-Expression (NFA) with specific Separation Patterns

Question: Given the RE (or NFA) for the set of all strings over $\Sigma ={a,b}$ such that: a occurs the odd number of times and each pair of a are separated by exactly $2n+2,n\geq 0$ b's. Attempt: ...
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NP-completeness of disjoint paths with bounded common nodes [duplicate]

Given an undirected graph $G=(V,E)$, $k$ distinct node pairs $(s_1, t_1), ..., (s_k, t_k)$ and an integer $\delta$, determine if there exist $k$ edge-disjoint paths from $s_i$ to $t_i$ $(1\leq i\leq k)...
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1answer
63 views

Sub-exponential time algorithm to compute playoff chances

There are 10 teams, Team A through Team J, playing in a triple round robin pool (each team plays thrice against each other team, for a total of a 27 games per team). After the round robin pool, the ...
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Which of the following statement is true?

Given two statement: A:if inputs of a problem is online, then there is faster or equal algorithm in view of time complexity for that problem when input is offline. B:if inputs of a problem is offline, ...
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1answer
36 views

A problem about asymptotic functions

Are there two function $f:N\rightarrow N$, and $g:N\rightarrow N$ such that $f(n)+g(n)\ne O(f(n))$ $\wedge$ $f(n)+g(n)\ne O(g(n))$? My idea: i think because of for any $f:N\rightarrow N$, and $g:...
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1answer
55 views

Polynomial time verification of Graph Isomorphism problem

Using guess and check method, for two given graphs with the same number of nodes, a NTM selects a permutation of the node set and then checks if the edges are preserved. The nondeterministic selection ...
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Length of PCP Proofs

Suppose a language $L$ has a non-adaptive PCP verifier which uses at most $r(n)$ random coins, and queries at most $q(n)$ locations of the proof, on inputs of length $n$. How can we show that there is ...
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Lower bound $\Omega$ grows quicker than upper bound $O$ of a recurrence relation $T(n)$?

In my analysis of algorithms class we were given the following recurrence relation: \begin{eqnarray} T(n) &=& \begin{cases} T\left(\displaystyle\frac{n}{2}\right) + 1, &n \ \mbox{is ...
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Reduction between Parity-SAT and approximate counting

Consider two problems as defined here. Approximate counting: Given a Boolean function $f(x)$, for $x \in \{0, 1\}^{n}$, distinguish between the two cases: The number of satisfying assignments for $f(...
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Need help with a very specific Greedy Algorithm. Are there fast solutions?

i need help for a specific problem i have at work. You have N number of rows in an array, each with some distribution of Numbers that range from 0 to N.Given an array of size 1000: Row 1 might look ...
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1answer
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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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40 views

Is there a concept of probabilistic quantum computers?

Answering my question Yonatan N said a statement from which follows that there are computable functions of quantum time complexity strictly above polynomial. Accordingly a Quora answer Quantum ...
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60 views

Why do we use DAG rather than trees to represent search space of a search problem?

I saw people use DAGs to represent the search space of a search problems like the travelling salesman problem. Why is this better than the tree representation? Is the reason to save memory space on ...
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1answer
106 views

Efficient algorithm to compute the diameter of a convex set?

Is there a polynomial algorithm that can compute the diameter (the distance between the furthest points) of a convex set? It is possible to do it efficiently for a set of points, but imagine that the ...

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