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Questions tagged [cc.complexity-theory]

P versus NP and other resource-bounded computation.

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5
votes
1answer
190 views

Boolean circuits which correspond to L/poly

Branching programs are usually used as a computation model for non-uniform logarithmic space $\mathsf{L}/\mathrm{poly}$. Is there a reference about Boolean circuits corresponding to $\mathsf{L}/\...
4
votes
2answers
357 views

Complexity of finding a consistent hyperplane

Given $m$ binary labeled points in $\mathbb{R}^d$, it is well-known that in general it's NP-hard to find a hyperplane that minimizes sample error. A brute-force search considers all $O(m^d)$ sample ...
3
votes
1answer
103 views

Does a non-constructive proof of bounds of a computable asymptotic complexity, with impossible fix, exist?

Does there exist an algorithm, about which a non-constructive $\omega$-consistent theory $A$ can prove that it has time complexity $O(f(n))$ where $n$ is some univariate function of the input, but ...
8
votes
0answers
242 views

L/quasipoly vs NL/poly

Savitch's theorem shows that NSPACE($S(n)$) $\subseteq$ SPACE($S(n)^2$), which means that nondeterminism can be replaced by more spaces in this situation. Is it known whether nondeterminism can be ...
0
votes
1answer
438 views

What are the problems in EXPSPACE \ EXPTIME?

Based on the diagram below (from Papadimitriou's book) we can see that PSPACE ⊆ EXPTIME ⊆ EXPSPACE. We know that PSPACE ⊊ EXPSPACE, which means that there are problems that can be solved in polynomial ...
6
votes
0answers
96 views

Efficient quantum algorithm for CLASSICAL FFT

Is there a known improvement on the current O(n*log(n)) algorithm for CLASSICAL FFT using quantum computation? 'n' is the number of samples. I need to find the amplitude and phase of the K dominating ...
7
votes
1answer
214 views

TIME(n) versus TIME(nlogn)

The time hierarchy theorem implies TIME($n$) is strictly contained in TIME($n\log^{1+ε}n$) for all ε>0. Is the relationship between TIME($n$) and TIME($nlogn$) known?
7
votes
0answers
56 views

Complexity of deciding whether subspaces of Z_2^n cover every point 3*x times

When studying the complexity of checking identities in certain finite algebras, I came across the following decision problem: Input: A positive integer $n \in N$ and a set of affine subspaces $H_1,...
5
votes
0answers
104 views

Parsimonious Reduction from Unique-3SAT to NAE-3SAT

Using the result by Valiant and Vazirani, we know that Unique-3SAT (3SAT with a unique solution) is hard unless NP=RP. Also it is widely believed that the "Unique" version of any NP-complete problem ...
9
votes
1answer
334 views

“Verifiable information”: is this a known concept?

The following seems to me like a natural definition and I wonder whether it's been studied somewhere Consider $\mathsf{X} \subset 2^{\lbrace 0, 1 \rbrace^*}$ a set of languages. Then $K \subset \...
43
votes
8answers
6k views

Best Upper Bounds on SAT

In another thread, Joe Fitzsimons asked about "the best current lower bounds on 3SAT." I'd like to go the other way: What's the best current upper bounds on 3SAT? In other words, what is the time ...
1
vote
0answers
43 views

How to efficiently verify if a semantic symmetry of a CNF formula is valid?

It is easy to verify that a syntactic symmetry of a CNF formula is correct. Is it also possible to check in polynomial time that a semantic symmetry which is not a syntactic symmetry of a formula is ...
5
votes
1answer
109 views

Is Bayes optimal RL of a finite set of DFAs feasible?

Let $Q$ be a finite set of states, $\Sigma$ a finite alphabet, $q_0\in Q$ the start state and $F\subseteq Q$ the set of accepting sets. Let $\{\delta_k:Q\times\Sigma\rightarrow Q\}_{k=1}^n$ be a set ...
28
votes
5answers
4k views

Fast Reduction from RSA to SAT

Scott Aaronson's blog post today gave a list of interesting open problems/tasks in complexity. One in particular caught my attention: Build a public library of 3SAT instances, with as few variables ...
4
votes
3answers
178 views

Complexity of isotopy of embedded graphs

I am looking for previous work on the following problem: given two graphs embedded in the plane without crossing, determine if they are isotopic. By isotopic I mean that there is a continuous ...
4
votes
0answers
110 views

Asymptotic complexity of mass production

For a function $f:\{0,1\}^n \rightarrow \{0,1\}^m$, let $C(f)$ be the circuit complexity (for concreteness, constants and NOT gates are free, while 2-input AND gates cost 1). Let $k{\times}f : \{0,1\}...
-1
votes
4answers
2k views

Interesting variation to the subset sum problem

An interesting variation of the subset sum problem was presented to me by a friend from work: Given a set S of positive integers, of size n, and integers a and K, is there a subset R (of the set S) ...
10
votes
1answer
322 views

Sorting with an average of $\mathrm{lg}(n!)+o(n)$ comparisons

Is there a comparison-based sorting algorithm that uses an average of $\mathrm{lg}(n!)+o(n)$ comparisons? Existence of a worst-case $\mathrm{lg}(n!)+o(n)$ comparison algorithm is an open problem, but ...
10
votes
1answer
220 views

Fast classical simulation of quantum algorithms

Are there examples of cases where the classical simulation of a quantum algorithm for a problem outperforms the best previously known classical algorithm for this problem? "Outperforms" doesn't have ...
17
votes
1answer
639 views

Example demonstrating the power of non-deterministic circuits

A non-deterministic Boolean circuit has, in addition to the ordinary inputs $x = (x_1,\dots,x_n)$, a set of "non-deterministic" inputs $y=(y_1,\dots,y_m)$. A non-deterministic circuit $C$ accepts ...
1
vote
0answers
57 views

Reasoning about NP hardness of optimization problems with closed form functions as input

(This may not be a research level question per se. I can delete this question if the community thinks this way too) I am trying to understand how to reason about hardness of optimization problems ...
232
votes
11answers
116k views

Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
3
votes
1answer
254 views

Conesequences of $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$

Does $\forall k\in \mathbb N \space NP\not\subseteq TISP(poly(n),n^k)$ has any separation of classes or consequences? My main question is can use this to show that $P \neq NP$ or some thing useful ...
0
votes
1answer
54 views

Understanding the definition of a “restriction of a resolution derivation”

I am reading the paper An Introduction to Lower Bounds on Resolution Proof Systems. In its subsection 2.3 the author gives an inductive definition of the restriction of a resolution refutation $\pi$. ...
-2
votes
1answer
267 views

Why is it a mystery if PSPACE ?= EXPTIME?

It seems obvious to me that $PSPACE \neq EXPTIME$. I, however, do not believe that my seemingly obvious logic would not be picked up by more intelligent people if it was so simple, so I'm assuming ...
1
vote
1answer
268 views

Is the following optimization problem (another variant to a previous problem) NP-hard?

This problem is a following up question on this one. The only difference is the addition of $3^{rd}$ constraint "$\sum_{i} x_{ij} \le M$", where M is a constant number. This constraint essentially ...
16
votes
0answers
428 views

An algebra of complexity classes

A key feature of unrelativized computation is its composability out of smaller fragments, and to partially capture the composability, I came up with an algebra of fine-grained complexity classes. For ...
3
votes
1answer
92 views

NP completeness of Hamiltonian cycle for the family of *dual graphs* to plane, cubic, triply connected graphs?

It is well known that the Hamiltonian cycle problem is NP-complete on the family of planar, cubic and triply connected graphs: https://epubs.siam.org/doi/abs/10.1137/0205049 For a problem I'm ...
30
votes
2answers
807 views

Is there a polynomial time algorithm to determine if the span of a set of matrices contains a permutation matrix?

I would like to find a polynomial time algorithm that determines if the span of a given set of matrices contains a permutation matrix. If any one knows if this problem is of a different complexity ...
4
votes
1answer
239 views

Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?

Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?
10
votes
1answer
274 views

Is this game EXPSPACE-complete?

Let $M$ be a polynomial-time deterministic machine that can ask questions to some oracle $A$. Initially $A$ is empty but this is can be changed after a game that will be described below. Let $x$ be ...
6
votes
1answer
152 views

Does there exist a polynomial time algorithm to determine whether an equation is a consequence of (x+y)*(y+z)=(x*y)+(y*z)?

Consider the identity $C$ $$(x+y)\times(y+z)=(x\times y)+(y\times z).$$ Is the problem of determining whether an identity is a consequence of $C$ in universal algebra a polynomial time problem? This ...
5
votes
3answers
454 views

Sorting a programs instructions until it works

Lets say I have a computer program below. (define (factorial x) (if (= x 0) 1 (else (* x (factorial (- x 1))))) I then take each line of the ...
3
votes
0answers
55 views

Pseudodeterministically choosing elements from efficiently samplable distributions (or, the plausibility of a weak choice principle)

Suppose we have a poly-time samplable family of distribution. I.e., a family of distributions $D_n \subseteq \{0, 1\}^{\mathsf{poly}(n)}$ and an algorithm $S$ for which $D_n = (r \leftarrow^\$ \{0,1\}^...
11
votes
1answer
319 views

Ruzzo-Simon-Tompa oracle access mechanism

In a paper on relativizing logspace computations, Ladner and Lynch construct an oracle relative to which $\mathsf{NL} \nsubseteq \mathsf{P}$. There are some more pathological examples in this vein in ...
11
votes
0answers
235 views

Inapproximability of multiterminal cut

In the multiterminal cut the input is a graph $G$ and a subset $T$ of its vertices. The task is to remove the minimum number of edges from $G$ such that there is no path connecting any distinct ...
5
votes
1answer
238 views

Distributions which are intractable to sample from?

I'm looking for an interesting family of probability distributions $P$ that is intractable to efficiently sample from. I'm not sure what the right notion of intractable is, though I know the notion ...
1
vote
1answer
76 views

Can we replace deterministic part of alternative turing machine with some other equivalent machines?

I'm sorry if it is a low level question but I am so confusing. If $DTime(n)\subseteq \Sigma_2Time(n^{0.2})$ then $DTime(n) \subseteq \Sigma_2DTime(n^{0.2})$ Is this true that $\Sigma_2DTime(n^{0.2})...
3
votes
0answers
219 views

Complexity of the mandelbrot set on rationals

(Also posted on mathoverflow) Given two rationals $a,b \in \mathbb{Q}$, call $c = a + ib$, i.e., the complex number represented by these two rationals. A point $c$ is contained within the Mandelbrot ...
4
votes
0answers
114 views

A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
2
votes
0answers
304 views

Relationship of the P=PSPACE problem to alternating complexity classes

From a very naive point of view of someone who haven't studied complexity theory in depth, I am wondering whether the theorems that APSPACE=EXPTIME and AP=PSPACE have any derived results on separating ...
3
votes
1answer
240 views

Can we efficiently distinguish between P and BPP?

Let's say algorithm $D$ distinguishes $BPP$ from $P$ if there exists a language $L \in BPP$ such that for all $A \in PTM$, $$D(\langle A\rangle) \in L \leftrightarrow D(\langle A \rangle) \notin L_A$$...
2
votes
0answers
197 views

Primality in $NC$ hierarchy?

AKS primality testing solves whether a given integer is prime in $P$. AKS algorithm is following: Input: integer n > 1. Check if $n$ is a perfect power: if $n = a^b$ for integers $a > 1$ and $b &...
2
votes
1answer
525 views

What is best known space requrement for solving SATISFIABILITY problem in exp time

I searched a lot for finding best space requirement algorithm for SATISFIABILITY problem but I didn't find any thing better than brute force that is in DSPACE(n). is there exists better bound? and ...
19
votes
1answer
532 views

Problems in BQP but conjectured to be outside P

Wikipedia listed four problems that are in $BQP$ but conjectured to be outside $P$: Integer factorization; Discrete logarithm; Simulation of quantum systems; Computing the Jones polynomial at certain ...
14
votes
1answer
1k views

Deciding whether an interval contains a prime number

What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
5
votes
2answers
176 views

Max cut problem between two connected subgraphs

Let $G$ be a connected graph. Consider the problem of finding a partition $G = A \cup B$ into connected subgraphs, so that the cut between $A$ and $B$ is maximized. Is there anything which is known ...
5
votes
1answer
140 views

What does “hold uniformly” mean in the context of asymptotic analysis?

What does "hold uniformly" mean in the statement of Theorem 1.7 in A Faster Subquadratic Algorithm for Finding Outlier Correlations? Here's the theorem text ("hold uniformly" is in the last line):
-1
votes
1answer
72 views

Restricted Universe Exact Cover

Apologies for a simple question - I am a beginning graduate student in TCS. Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and ...
2
votes
0answers
44 views

Min cut problem on unbalanced partitions for planar graphs with unit capacity edges

The question is: given a planar graph $G$ with unit capacity edge weights and a fixed positive integer $k$, what is an approximation algorithm for finding the minimum size of a cut $(A,B)$ with $|A|=k$...