Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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1answer
23 views

“Almost all objects have property P” vs. “It is easy to test whether an object has property P”

I am interested in any relation between "almost all objects(from a universe) possessing a particular property P" versus "testing whether an object has property P being poly. time decidable". My guess ...
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11 views

Is 1-in-3 SAT (exactly one 3-sat, or XSAT) #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles" by Moore and Robson, they prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat. Cubic Planar ...
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24 views

Randomized Reduction for Maximization Problem

I have two maximization problems $P_1$ and $P_2$ where the decision version $L_1 = \{(x, t) : \operatorname{Val}_1(x)\ge t\}$ of $P_1$ is $\mathsf{NP}$-complete. Let $f:P_1\to P_2$ be a randomized ...
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42 views

Are there problems that can be solved in time $2^{n/q}$ with $q$ qubits?

This is another attempt to formalize my former question on the topic. I'm looking for a problem for which all known classical algorithms take exponential time, but given ANY number of few qubits (...
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0answers
51 views

Violating the hierarchy theorems mildly

We know $NC\subsetneq PSPACE$ unconditionally. Do we know $PSPACE\subseteq NC/log$ is possible? Do we know $PSPACE\subsetneq NC/log$ is possible? Likewise take two classes $A\subseteq P$ and $B\...
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5answers
3k views

Implications of unprovability of $P\neq NP$

I was reading "Is P Versus NP Formally Independent?" but I got puzzled. It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is not ...
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1answer
160 views

Does two-sided error have more capability than one-sided error?

From $P=RP$ extrapolation we might think $EXP=REXP$. What evidence do we have $BPP\subseteq REXP$? What consequence $REXP\subseteq BPP$ gives other than what $EXP\subseteq BPP$ gives?
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How can I prove that a specific structure $\Pi_2$ problem is $\Pi_2$-complete?

The polynomial hierarchy problem is describled as following: $$\varphi_1:= \forall x_1\forall x_2\cdots \forall x_n\exists x_{n+1}\exists x_{n+2}\cdots\exists x_{n+m}~\bigwedge_{i=1}^n[f_i(x_{n+1},\...
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0answers
83 views

How fast can we perform DFT on a Log-RAM computer?

The question is how fast we can perform an $n$ coefficient discrete Fourier transform on the log-RAM model of computation. If we assume that we can do arithmetic operations on registers of size $\log{...
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79 views

Isomorphic subforest problem

I recently read that the following problem is NP-Complete: Given a tree $T$, and a forest $F$, is there a subgraph of $T$ isomorphic to $F$? The reference provide was to the textbook “Computers and ...
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Witness verifiable quantum advantage

As far as I can see, a major issue with Google's recent quantum supremacy claim is that it is hard to verify the results. If the quantum computer would be powerful enough to solve problems in NP (like ...
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1answer
862 views

Consequences of $\oplus \mathbf{P} \subseteq \mathbf{NP}$

I have part of a proof attempt of $\oplus \mathbf{P} \subseteq \mathbf{NP}$. The proof attempt consists of a Karp reduction from the $\oplus \mathbf{P}$-complete problem $\oplus$3-REGULAR VERTEX COVER ...
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1answer
358 views

Quantum Money where not even the Bank can counterfeit

The Quantum Money system proposed in "Quantum Copy-Protection and Quantum Money" has the following properties: The bank can produce bank notes in the form of quantum states. Anyone can verify that ...
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0answers
50 views

Can $\mathsf{P}^{\#\mathsf{P}}$ be described in terms of a non-deterministic (alternating) Turing machine?

Can the $\mathsf{P}^{\#\mathsf{P}}$ (= $\mathsf{P}^{\mathsf{PP}}$) class be described in terms of a non-deterministic Turing machine (in particular, an alternating Turing machine)? And would a $\...
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1answer
70 views

The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
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2answers
284 views

Bivariate low-degree polynomial testing of Polishchuk-Spielman

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
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1answer
47 views

Maximum subgraph problem with unknown complexity

Let $Q$ be a polynomial time decidable graph property. In a graph let us call a subgraph $S$ a $Q$-subgraph, if $S$ has the property $Q$. Consider the following optimization problem: Maximum $Q$-...
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2answers
342 views

The complexity of computing the permanent of a matrix of zeroes and ones versus a matrix of integers

How much easier is computing the permanent of a matrix with only zeroes and ones than a matrix of only integers?
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22answers
4k views

What hierarchies and/or hierarchy theorems do you know?

I am currently writing a survey on hierarchy theorems on TCS. Searching for related papers I noticed that hierarchy is a fundamendal concept not only in TCS and mathematics, but in numerous sciences, ...
9
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2answers
111 views

Complexity of finding an edge set yielding specified vertex degrees

I'm trying to figure out if the following two problems are known in general to be in P or NP-complete: Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'...
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1answer
147 views

What is the complexity of checking equivalence of two boolean formulae without NOT symbol?

Suppose I have two boolean formulae (propositions) $P_1$, and $P_2$ (can be assumed to be in CNF) over the same variables and such that there are no "NOT" symbols used. I.e. only conjunction and ...
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0answers
121 views

Terminology: FNP, with P replaced by NP?

Consider these two classes of search problems: Search problems with poly-sized solutions s.t. verifying solutions is in P. Search problems with poly-sized solutions s.t. verifying solutions is in NP. ...
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0answers
96 views

Matrix multiplication when one matrix is fixed

Let $A$ be a fixed positive entried integer matrix of size $a\times n$ with $\ell$ bits per entry One is allowed to pre-process this matrix as appropriate. Given another positive integer entried $B$...
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0answers
128 views

What are some examples of algorithmic applications of noncommutative rational identity testing?

The problem of polynomial identity testing (PIT) is known to be in $\mathsf{RP}$, but not known to be in $\mathsf{P}$. The related problem of noncommutative rational identity testing (NCIT) is known ...
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2answers
613 views

PPAD and Quantum

Today in New York and all over the world Christos Papadimitriou's birthday is celebrated. This is a good opportunity to ask about the relations between Christos' complexity class PPAD (and his other ...
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1answer
263 views

Consequences of BPP=BQP

If BPP=BQP then there is a polynomial time randomized factoring algorithm. A lot of other quantum algorithms that appeared to have an exponential speedup have recently been dequantized. For examples, ...
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2answers
678 views

What evidence do we have for (and against) Unique Games Conjecture?

Subhash Khot's Unique Games Conjecture is one of active research areas in complexity theory. What evidence do we have for it? What evidence do we have against it?
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4answers
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What evidence is there that Graph Isomorphism is not in $P$?

Motivated by Fortnow's comment on my post, Evidence that Graph Isomorphism problem is not $NP$-complete, and by the fact that $GI$ is a prime candidate for $NP$-intermediate problem (not $NP$-complete ...
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63 views

Time complexity of alternation free quantified linear program with no free variables and only existential quantifications

We know $\exists x\in\mathbb R^n:Ax\leq b$ is standard linear program. I am mainly looking at following case of quantified linear program with no free variables with only existential quantifications ...
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3answers
1k views

Is this variation of TQBF still PSPACE-complete?

Deciding if a quantified boolean formula such as $\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$ always evaluates to true is a classical PSPACE-complete ...
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1answer
268 views

Complexity classes not closed under intersection and union

Some of the better known complexity classes: PP, NP, P... are closed under intersection and union. What are some counter-examples? Is there a natural reason for the common complexity classes to be ...
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1answer
586 views

Consequences of sub-exponential proofs/algorithms for SAT

Would there be any major consequences if SAT had at most subexponential unsat proofs or even more strongly, SAT had subexponential-time algorithms?
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1answer
295 views

Which complexity class does this problem belong to?

Consider the following problem $\mathcal{P}$. Instance: A Boolean formula $F$ of $n$ Boolean variables ($x_1,...,x_n$) and $m$ Boolean parameters ($b_1,...,b_m$) where $0 \leq m \leq n$. Problem: ...
5
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1answer
91 views

Is the isomorphism problem between posets represented by DAGs GI-complete?

Given two directed acyclic graphs, how hard is the problem of checking whether the partial orders they represent are isomorphic? Is this problem GI-complete? I believe this problem is equivalent to ...
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1answer
198 views

Qubit gates in google supremacy

The gates in quantum supremacy experiment are nearest-neighbor and have spatial locality. Would this additional information help bolster IBM's argument to perhaps simulate quantum supremacy experiment ...
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6answers
2k views

Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
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1answer
177 views

Document references describing weaknesses for cutting planes and algebraic proof system?

Here, Fortnow says (section 4.3): Since then complexity theorists have shown similar weaknesses in a number of other proof systems including cutting planes, algebraic proof systems based on ...
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0answers
296 views

Is unary $\Pi_2$-SUBSETSUM coNP-complete?

Consider the following problem: for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation define is it true that for every $S \subseteq \{1, ..., 2n \}$ such that $|...
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1answer
115 views

How exactly is solving the random circuit sampling problem a computation in the Church-Turing thesis sense? [closed]

Note: This has been cross-posted to Quantum Computing SE. If we assume $\mathsf{BQP} \neq \mathsf{BPP}$, then we can say with reasonable certainty that Google's random sampling experiment falsifies ...
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1answer
224 views

What are the consequences of a faster algorithm for $CIRCUIT$-$SAT$?

What is the best algorithm known for $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates? What is the consequence if there is an $\alpha\in(0,1)$ such that $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates ...
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1answer
392 views

What would signify hierarchy collapse to first level?

We know that $$\mathsf{NP\subseteq P/Poly \iff coNP\subseteq P/Poly\implies PH=\Sigma_2^P}=\mathsf{\Pi_2^P}$$ $$\mathsf{NP\subseteq P/Log\iff coNP\subseteq P/Log\implies PH=\Sigma_0^P=\Pi_0^P}$$ ...
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1answer
289 views

Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses?

We think that $\mathsf{PH}$ does not collapse, and that $\mathsf{PP}$ is not in $\mathsf{P}$. Suppose on the contrary that $\mathsf{PH}$ does collapse, say even $\mathsf{P}= \mathsf{NP}$. $\mathsf{...
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2answers
215 views

Is the unbounded fan-in model realistic?

Does the unbounded fan-in circuit model apply in "practical" settings? In other words, are there real-world realisable computers with unbounded fan-in gates? As I understand, standard silicon ASICs ...
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10answers
5k views

Most important new papers in computational complexity

We often hear about classic research and publications in the field of computational complexity (Turing, Cook, Karp, Hartmanis, Razborov etc). I was wondering if there are recently published papers ...
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1answer
704 views

Technical issue with PCP theorem proof

I am reading the proof from here and I stumbled upon a technical (yet crucial) problem. I know this is rather specific and the context is problematic, but I couldn't figure it out myself. In pages 51 ...
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1answer
159 views

Arranging letters to make a word in a regular language

Fix a regular language $L$ on the alphabet $\{a, b\}$, and consider the following problem. I am given as input: some number $m \in \mathbb{N}$ of copies of the letter $a$, and some number $n \in \...
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1answer
69 views

Are there common names for the subtiers of PTIME?

We all know P, or PTIME, I think, as a common name for the class of polynomial-time problems. Are there common names for the first few levels inside P; that is, for constant-time, linear-time, ...
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1answer
156 views

On complexity class $\mathsf{\Pi_2 L}$

I suggest the following definition of $\mathsf{\Pi_2 L}$ (similarly to the certificate definition of $\mathsf{NL}$): A language $L$ belongs to $\mathsf{\Pi_2 L}$ iff there exists a deterministic ...
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0answers
52 views

Complexity of Block Design?

What is known about the complexity of creating Block Designs (https://en.wikipedia.org/wiki/Block_design)? I've found one paper that creates approximately solutions using Metaheuristics that claims ...
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57 views

Information theory for Mathematical Physics [duplicate]

What are some good introductory texts on information theory for someone who is classically trained in mathematical physics? Unfortunately my abilities in computer sciences and formal logic are next ...