# Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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### How do we show directly coNP is in MIP?

I know one can show that by $\mathsf{coNP}\subseteq\mathsf{NEXP}=\mathsf{MIP}$. But here I would like to start with a $\mathsf{coNP}$-complete problem and show there is a two-prover one-round ...
296 views

### What is the general consensus on the NL vs P question?

As it stands, we know that NL is a subset of P, but we do not know if it is a proper subset or not. With that said, what is the general consensus? Do the majority think that NL = P or not? What would ...
1 vote
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### Explanation of Complexity class $S_2^P$?

I am trying to understand the complexity class $S_2^P$ defined as (https://en.wikipedia.org/wiki/S2P_(complexity)). A language L is in $S_2^P$ if there exists a polynomial-time predicate $P$ such that:...
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### Why is the "balanced vs constant function" problem not a proof that P ≠ BPP?

Recently, I came across the problem of figuring out whether a given binary function $f(x)$ is constant (0 for all values of $x$ or 1 for all values of $x$) or balanced (0 for half of values and 1 for ...
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### Fine-grained average-case derandomization

Many believe derandomization with polynomial overhead, $\mathsf{P} = \mathsf{BPP}$, because it follows from $2^{\Omega(n)}$ circuit lower bounds for $\mathsf{E}$ (IW97). Do we have any evidence for or ...
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### Priority queue implementation with both find-min and delete-min $o(\log n)$

Question: There are several priority queue implementations listed on Wikipedia, along with amortized complexities of each of their basic operations: Does anyone know of an implementation in which the ...
1 vote
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### Oracle for the permanent-of-gaussians problem

In this paper, Aaronson and Arkhipov formulate the $GPE_\times$ problem as follows: given an $n \times n$ matrix $X$ of i.i.d. Gaussian random numbers, find the permanent of $X$ up to multiplicative ...
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### Baker–Gill–Solovay Theorem: why $2^n/10$ steps?

Context I'm teaching an introductory complexity theory course right now and although I work in adjacent areas, I'm not an expert on complexity theory myself, so I'm still in the process of working ...
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### Complexity class of optimization problems whose fractional relaxation is polynomial-time solvable

It is known that the problem of integer linear programming is NP-hard, but its fractional relaxation can be solved in polynomial time. The concept of fractional relaxation can be applied to any ...
1 vote
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### Variants of complexity classes that allow "adversarial inputs"?

Wikipedia defines BPP as follows: Alternatively, BPP can be defined using only deterministic Turing machines. A language L is in BPP if and only if there exists a polynomial p and deterministic ...
250 views

### Permutation generation problem using swaps

This is motivated by Aaronson's post, Probability of generating a desired permutation by random swaps. I am interested in a related problem where the swaps are given in the input. We're given as input ...
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### Is SZK dependent on the verifier’s model of computation?

What if instead of a probabilistic TM, the verifier in the definition of SZK was a quantum TM? How would this affect its relation to other classes? Would Statistical Difference still be a complete ...
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### Exchange cards with sum requirement

Given are positive integers $a_1,\dots,a_{2k},b_1,\dots,b_{2k},S$ such that $\sum_{i=1}^ka_i = \sum_{i=k+1}^{2k}a_i = S$ and $\sum_{i=1}^kb_i = \sum_{i=k+1}^{2k}b_i = S$. There are $2k$ cards, card $i$...
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### Open Quantum Analogs to Classical Problems

I am looking for interesting examples of complexity-theoretic and cryptographic problems where we have a significant amount of knowledge about the classical version of the problem, but we have no ...
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### Examples for Real-time vs Linear time

A real-time Turing machine (with multiple tapes) runs in linear time. It is known  that there are languages recognizable in linear time by a multitape Turing machine but not recognizable in real-...
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### Complexity of XOR-Knapsack

Edit: Actually I should have been more careful. Maybe the optimal way to solve this is to approach it as a series of $k'-$XOR sum problems (Generalized birthday due to Wagner) for increasing $k'.$ And ...
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### Reduction of Monotone-1-in-3-SAT to Cubic-Monotone-1-in-3-SAT

3-SAT is an NP-Complete problem. Now given a 3-SAT instance it can be transformed to a Monotone-1-in-3 SAT instance thus even Monotone-1-in-3-SAT is NP-Complete (am aware of this reduction). But, as I ...
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### Is it known that P $\neq$ NP implies BQP $\neq$ NP?

Pretty much the title. Is there any result that shows that $P \neq NP \Rightarrow BQP \neq NP$. I think it's pretty clear that $BQP \neq NP \Rightarrow P \neq NP$, as $P$ is a subclass of $BQP$. But ...
154 views

### Consequences of $P^{NP[o(n)]} = P^{NP}$

I am wondering what the consequences of $\text{P}^{\text{NP}[o(n)]} = \text{P}^{\text{NP}}$ are. Does this imply the collapse of the polynomial hierarchy or contradict something like $\text{ETH}$? I ...
5k views

### Theoretical Computer Science vs other Sciences?

So I‘m in my fifth semester studying Computer Science at a German university, so I‘ve only scratched the surface of Theoretical Computer Science, namely Logic, Formal Languages, Automata Theory, ...
193 views

### NP-hardness: (planar) directed feedback vertex set problem with bounded degree

My question is the directed version of this one. (I know the results and proofs about feedback vertex set in undirected graphs or undirected planar graphs; so I am concern about the directed feedback ...
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### $\mathsf{coNP}^{\mathsf{\#P}}$ and $\mathsf{coNP}^{\mathsf{\#P}^\mathsf{\#P}}$

I was reading a paper that demonstrates that deciding whether a loop-free program is $\varepsilon$-differentially private is $\mathsf{coNP}^{\mathsf{\#P}}$-complete. What are some other problems that ...
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### How is inapproximability by polynomial size circuits sufficient for the Nisan-Wigderson generator?

I couldn't understand how exactly Yao's XOR lemma was used to prove the following claim made in the proof of Theorem 2 of the original paper describing the Nisan-Wigderson generator, so I decided to ...
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### Sources that prove solving 2-SAT with DP takes linear time

Would anyone have any sources that describe/an explanation of how solving 2-SAT using dynamic programming takes a linear amount of time? Can't seem to find a text that proves it in detail/formality. ...
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### Complexity of checking if a given prime number can be computed using at most $s$ addition/multiplication operations?

Given are a prime number $p$ and a parameter $s\in\mathbb{N}$. What is the computational complexity of the problem of determining whether $p$ is computable by a series of at most $s$ steps, each being ... 90 views

### END OF THE LINE problem finding a node with in-degree $0$ or out-degree $0$ depending on the initial node

The END OF THE LINE problem is stated as Given two circuits, P and N, a node, v, is balanced if $P(N(v)) = N(P(v)) = v$ or $P(N(v)) \neq N(P(v)) \neq v$. Given that $0^n$ is not balanced, find ...
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### Lower bound for sorting without using a decision tree model

Can we prove the lower bound for the sorting problem just by Turing machine model? It seems that available proof of sorting is based on the assumption that the algorithm only uses comparison so we can ...
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### NC0 randomnes vs. non-uniformity

In Ajtai and Ben-Or. A theorem on probabilistic constant depth Computations. STOC '84, 1984 Ajtai and Ben-Or show a non-uniform derandomization of BPAC0. Is there a similar relation known for ...
152 views

### What is the simplest one-way function (in terms of boolean circuit complexity)?

What is the simplest known one-way function? By simplest, I mean, when implemented as boolean logic, the number of AND/OR/NOT gates needed is minimal (smallest circuit complexity). (I'm trying to find ...
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### Is it $NP$-hard to check whether a given algebraic circuit computes permanent?

Given are a natural number $n\in\mathbb{N}$ and a polynomial $P$ in the form of an arithmetic circuit $C$ over $\mathbb{Z}$ (a circuit which only uses $+$ and $\times$ gates and integer constants as ... 113 views

### Complexity of the unique homomorphism problem up to automorphisms

I am interested in the following problem: given two relational structures $\mathbf{A},\mathbf{B}$, is there a unique homomorphism from $\mathbf{A}$ to $\mathbf{B}$ up to automorphisms of $\mathbf{B}$, ...
1 vote