Questions related to the (computational) complexity of solving problems

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3
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Intuition for PH notation in Arora-Barak's Computational Complexity

For the definition of polynomial hierarchy: $x \in L \Leftrightarrow \exists u_1 \in \{0, 1\}^{q(|x|)}\forall u_2 \in \{0, 1\}^{q(|x|)} \cdots Q_i u_i \in \{0, 1\}^{q(|x|)} M(x, u_1, \ldots, u_i) = ...
2
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1answer
14 views

Ford-Fulkerson Running Time

This question might be really basic but every source seems to skip over a couple of steps neither of which seem trivial to me. It would be great if someone could explain them! In the analysis of ...
1
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1answer
57 views

Can Circuit Value Problem or HORN-SAT be reduced to PATH problem?

PATH = {(X,R,S,T) | exists an x in S that is admissible} Where R is a relation of X x X x X, S is a unary relation of X and T is a unary relation of X aswell. An x element of X is admissible if it is ...
1
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2answers
55 views

How important is it to find a deterministic polynomial time algorithm to construct Ramanujan graphs? [on hold]

As in I don't know what is the difference between say the conferences SODA, STOC or FOCS. Measured in terms of such conferences, where would such a result be publishable? This is not a "technical" ...
4
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1answer
39 views

What usage is the delta defined in the polynomial hierarchy?

At the Wikipedia page, the polynomial hierarchy also defines the following: $\Delta_0^\text{P} = P$, $\Delta_i^\text{P} = \text{P}^{\Sigma_{i-1}^\text{P}}$ However, the only usage of this anywhere ...
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0answers
12 views

Arthur-Merlin protocol to decide a set size

Please look at the example here at the bottom of page 3, http://www.cs.nyu.edu/~khot/CSCI-GA.3350-001-2014/sol3.pdf Here it seems that the set whose size Arthur is trying to approximate is known in ...
1
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1answer
34 views

About a particular use of hashing

Look at the last problem on page 2 here, http://www.cs.nyu.edu/~khot/CSCI-GA.3350-001-2014/sol3.pdf All one wants to do is to convert a $x \in \{ 0,1\}^n$ into a $y \in \{0,1\}^k$ . Then just a ...
3
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1answer
44 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 ...
5
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1answer
59 views

TM recognizing $0^n1^n$ requires Ω(log n) space

I am trying to prove that any deterministic 1-tape Turing Machine which recognizes the language $L = \lbrace{0^n1^n | n \geq 0 \rbrace}$ requires $\Omega(\text{log }n)$ space. I believe this can be ...
-2
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1answer
28 views

Is a single tape Turing machine equal in power to a Turing machine that can only move right? [duplicate]

I assume no, because a Turing machine that can only move right feels like it is not a Turing machine. But, I wonder if I can add a Reset to the right moving Turing machine that resets the what head ...
3
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1answer
378 views

Good introduction to Turing's work and complexity theory?

I'm currently an undergrad whose been amazed by what Turing has done for the world. I know there are plenty of other amazing individuals, but Turing's work specifically has always sounded the most ...
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 ...
3
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1answer
23 views

How to state that a complexity bound does not depend on a given parameter size?

I am often ill at ease with Landau (Big O) notation, because it seems often to be abusing mathematical notation. The best example is the use of the equal sign to express a set membership. And this can ...
2
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1answer
110 views

How to improve my these specific math skills? [closed]

I am student of CS. Problem is, I feel that I don't have enough math knowledge to solve mathematical problems. When some programming problems arises which needs some math skills to solve then despite ...
6
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1answer
70 views

Is there a continuous hash?

Questions: Can there be a (cryptographically secure) hash that preserves the information topology of $\{0,1\}^{*}$? Can we add an efficiently computable closeness predicate which given $h_k(x)$ and ...
0
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0answers
13 views

More about the ESP tree

In this previous question I had asked about the intuition behind looking at the ESP tree. One place where it is used is to construct an approximation of arbitrary distance functions $d : [m]^n ...
2
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1answer
26 views

How to read $NC^1\subset L \subset NL \subset SAC^1$, $SAC^1=LOGCFL/poly$, and similar statements?

The (complexity zoo) description of $NC^1$ says that it is contained in $L$, i.e. $NC^1\subset L$. The description of $SAC^1$ says that it is equal to $LOGCFL$$/poly$, i.e. $SAC^1=LOGCFL/poly$. The ...
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1answer
25 views

Use Rice's theorem to prove the following is undecidable

Given the language $L=\{\alpha \mid M_{\alpha}(x)=x^3$ for all $x\in\{0,1\}^*\}$. Prove using Rice's theorem that $L$ is undecidable. Rice's theorem: Let $P$ be a set of all computable functions ...
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0answers
19 views

How to formally show that $f(n) = o(g(n))$ [duplicate]

Given $f(n)=n^{100}$ and $g(n)=2^{\frac{n}{100}}$, I can tell that $f(n)=o(g(n))$ mainly because $n^c=o(2^n), \forall c$. But how can I formally show that $f(n)=o(g(n))$. My attempt at this question ...
1
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1answer
33 views

Complexity terminology

What is the terminology used for speaking about complexity, when we don't study it asympotically (but exactly) ? Thank you
2
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1answer
29 views

Is it possible that low-resource Turing Machines can always “usually” agree with high-resource Turing Machines

Say that a language $L$ is a $f$-approximation of a language $L'$ if, for all input lengths $n$, $L$ and $L'$ agree on at least a fraction $f$ of the inputs. It is known that there are problems in ...
1
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1answer
30 views

Can well-formed formulas in predicate logic for a given signature be recognized in LOGSPACE?

I read that visibly pushdown languages are supposed to model the typical simple formal languages like XML better than deterministic context free languages. The visibly pushdown languages can be ...
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 ...
2
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1answer
31 views

Probabilistic algorithm with two-sided error

I am currently studying probabilistic algorithms and came across three major complexity classes: BPP: worst-case polynomial time, two-sided error RP: worst-case polynomial time, one-sided error ZPP: ...
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0answers
8 views

Removing the acceptance error from AM

Typically the AM class is defined with error upper bound of 1/3 for deciding both the situations of the membership question being true or false. But curiously enough for the situations when the ...
2
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1answer
46 views

Cycles in hardness of ST-CON for the class NL

It seems to me that the problem of $s$, $t$ connectivity in a DAG should still be NL-Complete. I am aware that ST-CON without the DAG restriction is complete for NL, so obviously the DAG restriction ...
3
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1answer
39 views

Proving NP completness without reductions

What methods are there to prove a language is NP-complete? I already know the reduction method, but are there more sophisticated/advanced methods to prove this?
1
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1answer
19 views

How does derandomization of 3SAT work via conditional expectations?

Given a single SAT clause with its 3 literals coming from 3 different variables it is obvious that a random assignment of values will satisfy it with probability 7/8 But I do not understand how ...
3
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1answer
30 views

What is the rigorous definition of an efficient algorithm that $\epsilon-refutes$ random 3CNF formulas

I recently asked a similar What does "refuting random 3CNF" formulas mean?, however, I'd like to address it in a more mathematically precise setting. In that paper, on page 5, it talks ...
5
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1answer
41 views

bounded length CoNP proof

Question: Let $A \subseteq $ {0,1}$^* $ be a language which satisfies $|A \cap ${0,1}$^n|=n^3 $ for all $n\ge 10$ Prove that $A \in NP$ implies $A \in coNP$. Thoughts I've been having difficulty ...
1
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1answer
31 views

Are there any theorems/formulas that apply to the height of comparison trees?

I have been drawing some binary comparison trees, which correspond to compares made to sort an array, and I was wondering if there is any formula to determine the height of a comparison tree for an ...
3
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2answers
62 views

What does “refuting random 3CNF” formulas mean?

Intuitively, recall what 3CNF formulas mean: Its a boolean formula with conjunctive normal form (i.e. formula of ANDs of clauses with ORs) with no more than three variables per conjunct. I was ...
1
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1answer
18 views

Are there two kinds of polynomial hierarchy collapses?

It seems to me that there are two different situations which get called ``PH collapse", (1) That $\exists i \geq 1$ s.t $\Sigma_i ^p = \Sigma_{i+1}^p$ (2) That $\exists i \geq 1$ s.t $\Sigma_i^p = ...
5
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1answer
31 views

Hardness of a problem related to set cover

Suppose $C_1, \ldots, C_m$ are subsets of $\{1, \ldots, n\}$. The goal is to find the smallest subcollection of $C_1, ..., C_m$ such that each element of $\{1, \ldots, n\}$ appears at least $k$ times ...
3
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1answer
26 views

What is the decidable language in $P/poly$ but not in $P$?

Except for the undecidable unaries I have no idea if there is anything in the gap between $P/poly$ and $P$
5
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0answers
45 views

Is the compuation of a minimal correction subset (MCS) $FP^{NP}$-hard? [migrated]

MCS problem: Given a set $\phi$ of Boolean clauses. Find a minimal correction subset (MCS) $M\subseteq\phi$ such that: $\phi\setminus M$ is satisfiable and for all $c\in M$ holds $\phi\setminus ...
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 ...
0
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0answers
15 views

Subset product problem for powers of 2

We know that the "subset product problem" is NP-complete, as Gary and Johnson mentioned in their book, and the proof is by reduction from X3C. I wonder if we can prove that this problem, i.e., the ...
3
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1answer
34 views

Lower bound for maxima on 2D plane

Given $n$ points $(x_1, y_1), \ldots, (x_n, y_n)$ on a 2-dimensional plane. A point $(x_1, y_1)$ dominates $(x_2, y_2)$ if $x_1 > x_2 \land y_1 > y_2$. A point is called a maxima if no ...
2
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1answer
21 views

How does one change the probability bounds in probabilistic complexity classes without changing the class?

I see this theorem whose proof is not clear to me : "Let $L \subseteq \{0,1\}^*$ be a language and suppose that there exists a polynomial time PTM M such that for every $x \in \{0,1\}^*$ and $Pr[ ...
2
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2answers
35 views

Difficult Cases for 3MaxSAT and 3SAT Approximation Algorithm

Its known that a polynomial time approximation algorithm that satisfies 3MaxSAT in 7/8+e clauses implies P=NP. Its also experimentally known that 3SAT has the most difficult known cases when the ...
2
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1answer
93 views

$P \neq NP$ and determinism

Suppose $P \neq NP$. Does it imply that there exists some superpolynomial time bound, such that any $NP$-complete problem, like SAT, can be used to simulate an arbitrary deterministc Turing Machine ...
2
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1answer
20 views

Sensitivity versus degree

Given boolean function $f$, let $F$ denote the unique multiaffine real polynomial representing $f$. Sensitivity of $f$ at input $x$ is $$S_x(f) = |\{i:f(x)\neq f(x^i)\}|$$ where $x^i=x\oplus\Bbb 1_i$ ...
5
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1answer
107 views

Does $P \neq NP$ imply $NP \neq PSPACE$?

Is it true that $\mathsf{P} \neq \mathsf{NP}$ implies $\mathsf{NP} \neq \mathsf{PSPACE}$? I have here some problem that is in $\mathsf{PSPACE} \setminus \mathsf{NP}$ if $\mathsf{P} \neq \mathsf{NP}$. ...
2
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1answer
26 views

Can one use the PCP theorem to prove correctness of deternimistic algorithms?

I am thinking of the equality "PCP(O(log(n)),0) = P" Say I have a deterministic polynomial time algorithm $A$ whose correctness I can't prove immediately. But say I create a probabilistic version of ...
4
votes
1answer
51 views

Difference between time complexity and computational complexity [duplicate]

For measuring the complexity of an algorithm, is it time complexity, or computational complexity? What is the difference between them? I used to calculate the maximum (worst) count of basic (most ...
8
votes
2answers
236 views

Difference between time complexity and computational complexity

For measuring the complexity of an algorithm, is it time complexity, or computational complexity? What is the difference between them? I used to calculate the maximum (worst) count of basic (most ...
1
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0answers
54 views

What is a good example of an NL-complete context free language?

Setting Exactly as the title stated: Give an example of an $\mathsf{NL}$-complete context free language. $\newcommand{\angle}[1]{\langle #1 \rangle}$ Current Solution Recall in the past we ...
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1answer
42 views

What are some of the best circuit separations that we know of and that we suspect?

We know that $\mathsf{non-uniformAC^0\subsetneq PSPACE}$, $\mathsf{non-uniformACC^0\subsetneq NEXP}$. We know that $\mathsf{uniformACC^0=PH}$ is a possibility. What are some of circuit separations ...