Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
2,658
questions
3
votes
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 ...
-2
votes
0answers
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
votes
0answers
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.
...
-1
votes
0answers
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
vote
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
votes
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
votes
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 ...
-1
votes
0answers
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
votes
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
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
11
votes
3answers
2k 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 ...
0
votes
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-...
0
votes
0answers
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
votes
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
votes
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?
2
votes
0answers
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
votes
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
votes
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
votes
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 ...
2
votes
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 $\...
1
vote
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
0
votes
0answers
20 views
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
votes
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
votes
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
votes
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'...
4
votes
0answers
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.
...
3
votes
0answers
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
votes
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, ...
1
vote
0answers
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 ...
9
votes
0answers
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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 ...
2
votes
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 ...
0
votes
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 ...
10
votes
0answers
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 ...
-1
votes
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
votes
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?$$
8
votes
0answers
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
votes
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
votes
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 $|...
