Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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3
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1answer
91 views

$DTIME_1(o(n^2))\setminus$ REGULAR

Maybe this is well-known, but I couldn't find any example of a non-regular lanugage that is decidable on a single-tape Turing machine in subquadratic time. Help! Related paper: On the structure of ...
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55 views

Weakest condition that connects $P=BPP$ and derandomizing $VV$

It is known $P=BPP$ is insufficient to derandomize $VV$ isolation lemma. What does it mean to be 'insufficient' here? Is there some theorem which says $P=BPP$ $+$ 'condition $A$' gives ...
9
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114 views

Alternative proofs of Savitch's theorem?

Question: Are there any known proofs of Savitch's theorem that $NL \subseteq L^2$ besides the usual one? By the usual one I mean the proof based on recursively querying whether there is a midpoint. ...
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27 views

Complexity of solving systems of linear equations with hash preimages

Consider the following problem: Let $H$ be an arbitrary cryptographic hash function. The goal is to solve the following system of linear equations with integer coefficients $$\boldsymbol A \...
1
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1answer
46 views

Matrix rank approximation

I am aware of the problem of low rank approximation of matrices which has been studied in various models of computation. My question is the following: What is the status of approximating rank of a ...
5
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1answer
109 views

Complexity of finding automorphism group of code

What is the computational complexity (may be both classical or quantum) for finding automorphism group of a general linear code? Is there better bound on complexity if structure of code is known for ...
2
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1answer
142 views

Are there problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not hard for $DTIME(n^{k-1})$ under nearly linear time reductions?

Background It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$. Although some natural problems are known to exist, many of them ...
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47 views

Fastest way for boolean matrix computations

I have a boolean matrix with 1.5 million rows and 20k columns, similar to this example: ...
2
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2answers
157 views

Problem in deterministic time $n^p$ and not lower

I'm looking for any language $L$ candiate to be in $DTIME(n^p) -DTIME(n^{p-1})$ (it takes at least $n^{p-1}$ steps to determine if an input is in L with a 2-tape $TM$, but L is polynomially solvable). ...
7
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1answer
114 views

The complexity of the permanent of low rank matrices

I know that for an arbitrary $n \times n$ matrix, Ryser's algorithm can compute the permanent in $\mathcal{O}(2^n n^2)$ time. I'm interested in computing the permanent of $n \times n$ matrices of rank ...
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1answer
65 views

the shorstest cycle containing two given points

I am given a edge-weighted (multi)graph $G$ and two of its vertices, $u, v\in V(G)$. I want to find two edge-disjoint paths that connects $u$ and $v$ while minimizing the sum of the lengths of the ...
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1answer
86 views

Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?

We conjecture that Hamiltonian cycle is fixed parameter tractable with parameter clique cover, given $k$-clique cover. Let $G$ be connected simple graph. $k$-clique cover of graph $G$ is partition ...
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3answers
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“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 ...
0
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1answer
85 views

Is the counting version of 1-in-3 Sat #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat. Cubic Planar Positive 1-in-...
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44 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 ...
4
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1answer
212 views

Are there problems that can be solved in time $2^{n-q^c}$ 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 (...
0
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1answer
172 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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88 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{...
3
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0answers
87 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 ...
4
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0answers
145 views

Witness verifiable quantum advantage

Update: A slightly different version of this question has been answered here. 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. ...
7
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1answer
374 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
52 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
72 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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Consequences of $P=RP$ to algebraic geometry

$P=RP$ gives $NP$ in $\oplus P$ and relativing gives $PH$ in $\oplus P$. Would it have any impact to algebraic geometry via analogue of Toda's result shown in https://arxiv.org/abs/0912.2652?
2
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1answer
49 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 $Q$-...
9
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1answer
149 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 ...
9
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2answers
112 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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123 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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108 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 $B$...
4
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1answer
275 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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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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130 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 ...
5
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1answer
92 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 ...
5
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1answer
208 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 ...
7
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1answer
180 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 ...
4
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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: ...
1
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1answer
116 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 ...
3
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1answer
275 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 ...
5
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1answer
228 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 ...
8
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3answers
505 views

Is the wording of Google's QC Supremacy valid?

Quantum supremacy using a programmable superconducting processor was published today. Scott Aaronson posted a few weeks ago a post about this paper and it was clear we will see a Nature or Science ...
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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, ...
4
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1answer
157 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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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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0answers
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 ...
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183 views

On Courcelle's question about Monadic second-order logic with cardinality predicates

I have found the following question at openproblemgarden.org: The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the ...
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1answer
58 views

Chomsky-Schutzenberg Hierarchies explained for physicist (general) [closed]

I am classically trained in physics, however I have been interested in the use of information theory in studying some classical systems. As someone who is somewhat unfamiliar with the language of ...
4
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1answer
192 views

Results comparing BQP and NEXP

Are there oracle results with $$P=NP\neq BQP=NEXP\mbox{ and }P=NP\neq BQP\neq NEXP?$$ Also is there a $PCP$ characterization of $BQP$ like $$PCP(O(poly(n)),1)=PCP(O(poly(n)),O(poly(n)))=NEXP?$$
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146 views

A canonical complete problem for EXP and NEXP in terms of formulae

3SAT is a complete problem for NP. TQBF is a complete problem for PSPACE. Is there direct way to define canonical complete problems for EXP and NEXP in terms of boolean formulae? I have only seen ...
6
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1answer
323 views

On theoretical aproaches for solving $\mathsf{SAT}$ in special cases

In what cases $\mathsf{SAT}$ can be solved in polynomial time? I know two cases: $2$-$\mathsf{SAT}$ and Horn-$\mathsf{SAT}$. Question 1: Is there a reference with algorithms for solving $\mathsf{...
10
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0answers
298 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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