Questions tagged [cc.complexity-theory]
P versus NP and other resource-bounded computation.
-3
votes
0answers
33 views
How can we solve the recurrence $T(n) = T(n/2) + log^2(n)$ [on hold]
I am trying to solve the following recurrence:
$T(n) = T(n/2) + log^2(n)$
So far I have come up with the following expression
$T(n) = \sum^{log(n)}_0 2^klog(n/2^k)$
I cannot figure out a way to ...
-4
votes
0answers
20 views
Prove that language in L
How to prove that $L = \{(\phi,x) | \space Boolean \space formula \space \phi \space is \space true \space on \space a \space set \space of \space x\} $ lies in L?
-5
votes
1answer
49 views
Can I prove this polynomial reduction?
Consider Barely-3-SAT: given a CNF $F=\wedge^m C_i$ determine whether there is an assignment such that exactly one clause $C_k$ is true all the others are false. I want to show this is $NP$-complete. ...
-3
votes
2answers
147 views
Do Turing complete languages automatically have efficient algorithms [on hold]
Every Turing complete programming language can describe an algorithm that sorts sequences. Is it also true that every Turing complete language can describe an algorithm that sorts sequences in $\...
12
votes
1answer
282 views
Is there a P-complete language X such that succinct-X is in P?
I came across a paper called "A Note on Succinct Representation of Graphs". It seems that in the discussion section they claim that for any problem $X$ that is $\mathrm{P}$-hard under projections, $\...
-3
votes
0answers
252 views
Is there any chance these results are correct? [closed]
P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P ...
4
votes
2answers
206 views
Is this partition problem strongly NP-complete?
Some computational problems have variants that appear to be harder. For instance, Graph Automorphism (GA) problem has quasi-polynomial time algorithm ( by Babai's Graph Isomorphism result) while the ...
2
votes
0answers
139 views
Best polynomial-time approximation factor for NP-optimization problems
Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold:
There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
-4
votes
0answers
74 views
Prove that if A is NP-complete and B is coNP-complete, than AxB is NP-, coNP-complete
AxB means cartesian product of A and B.
May someone help me with this? I even have no idea how to prove that AxB belongs to NP or coNP
8
votes
0answers
92 views
Does ${\bf CFLPAD}={\bf PPAD}$?
What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a (non-)deterministic finite/push-down automaton encodes the problem?
I asked a similar ...
0
votes
1answer
93 views
Permuting the columns of a 0/1-matrix to avoid short segments
Consider an $n \times n$ table with $n$ stars such that each row contains at most $\log n$ stars. The stars break each row into segments (continuous parts of a row without stars). Let's call a segment ...
5
votes
1answer
148 views
Minimal information needed for determine some function
From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
-4
votes
2answers
77 views
Is it possible to have a sorting algorithm that computes faster than QuickSort? [closed]
Given an unsorted array, QuickSort has to touch each source element it is trying to sort multiple times before it declares an array as sorted.
(notice how many times the 2 is touched [circled in red ...
6
votes
0answers
76 views
Can relativization technique be applied to natural NP-complete languages?
Levin [1] defined distNP is the distributional problem (L,D), where L ∈ NP, and D is an ensemble of efficiently samplable distributions over problem instances. We say that a distNP problem (L,D) is ...
1
vote
0answers
41 views
Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time? [closed]
In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
2
votes
1answer
198 views
Deciding whether a graph contains a complete balanced bipartite graph
Is it known whether the following problem is in P or is NP-complete?
Problem: given an input graph $G$ on $n$ vertices, decide whether $G$ contains a complete $n/2 \times n/2$ bipartite graph.
-1
votes
0answers
39 views
Proofs of the form “$A, B \in \mathcal{NP}\setminus{\mathcal{P}}, \ A, B \not\in \mathcal{NP-C}, \ A \not\leq_p B$” [migrated]
I'm currently learning about complexity classes and I have a few general questions:
Assume $A, B \in \mathcal{NP}\setminus{\mathcal{P}}, \ A, B \not\in \mathcal{NP-C}$
are people generally ...
3
votes
0answers
71 views
Fixed dimension Linear Integer Programming in $NC$
We know if fixed dimension linear integer programming is in $NC$ then integer $GCD$ is in $NC$. Is this the only non-trivial implication of fixed dimension linear integer programming in $NC$?
1
vote
0answers
196 views
EXPSPACE proof and its implications
I'm dealing with the min-max regret 0-1 Integer Linear Programming problem (MMR-ILP, for short), which is formulated as below.
\begin{equation} \label{eq:nip_obj}
\min_{x \in \Phi} \sum_{i = 1}^n ...
4
votes
1answer
135 views
Complexity of Acyclic Hypergraph Isomorphism
It is well known that the graph isomorphism problem restricted on trees is much easier than the general case. It can be done in logarithmic space (Jenner B, Lange KJ, McKenzie P. "Tree isomorphism and ...
1
vote
1answer
76 views
Is balanced Hamiltonian cycle NP complete on maximal plane graphs?
I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs.
If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
1
vote
0answers
59 views
Complexity of enumerating over promise problems and circuits?
Given an enumeration over all Turing Machine which run with increasing length, is there a ``complexity class'' which describes the complexity of determining whether a given TM satisfies the promise ...
11
votes
0answers
115 views
Can two-tape read-only Turing machines recognize any recursive language?
Suppose that a $k$-tape read-only Turing machine receives its input on each $k$ tapes.
It cannot write on the tapes, but it can move on them in both ways, even move off from the input.
So for example, ...
11
votes
1answer
234 views
Is DSPACE(n) = DSPACE(1.5n)?
From space-hierarchy theorem it is known that if $f$ is space-constructible then
DSPACE($2f(n)$) is not equal to DSPACE($f(n))$.
Here, by DSPACE($f(n))$ I mean the class of all problems that can ...
1
vote
1answer
381 views
Possible to do Complexity theory with only counting and Pigeonhole
Most of the proofs in the book Computational complexity by Barak and Arora seem to be Pigeonhole in disguise. What are some places in Complexity theory where counting and Pigeonhole was insufficient ...
5
votes
1answer
233 views
Is Murphy's Law of Complexity Theory consistent? What separations/collapses does it imply?
A decade ago I observed what I dub "Murphy's Law of Complexity Theory": whenever a new separation or collapse is discovered, the question is answered in the direction that makes $P\overset?=NP$ most ...
-2
votes
1answer
116 views
Why $PSPACE!=Dtime(2^n)$? [closed]
Why $PSPACE != Dtime(2^n)$? I can not see how padding argument can help here, how can it be proven?
9
votes
1answer
125 views
Best known asymptotic PCP sizes / 3-SAT
What are the best known asymptotic upper bounds on sizes of probabilistically checkable proofs? Ideally, I am looking for a contemporary survey on this broad question, but if there is none, I am ...
10
votes
0answers
94 views
Complexity of checking $a > br^m + cr^n$, with $r$ rational
I'm wondering if the following problem is decidable in P-time (or even NP):
Given $a, b, c \in \mathbb{Z}$ and $m, n, p, q \in \mathbb{N}$ all in binary, decide if $a > br^m + cr^n$, where $r = {p ...
2
votes
1answer
96 views
Minimization version of matrix p-norms?
I considered a minimization version of matrix p-norms, defined for a matrix $A$ by
$$
f_p(A)= \min_{x\neq 0} \frac{||Ax||_p}{||x||_p}.
$$
Notice that $f_p(A) = 0$ if and only if $A$'s columns are ...
1
vote
1answer
235 views
$P=BPP$ without good PRGs?
We know that the existence of good pseudorandom generators (PRGs) does not only imply $P=BPP$, but also $PromiseP=PromiseBPP$.
Let us assume $PromiseP\ne PromiseBPP$. Then good PRGs do not exist. ...
5
votes
1answer
105 views
Complexity of counting integer roots of multivariate polynomials in a polyhedron?
Deciding integer roots of multivariate polyomials is undecidable. However what is known about counting integer roots of multivariate polynomials in $\mathbb Z[x_1,\dots,x_m]$ with both $m$ and total ...
4
votes
1answer
64 views
Complexity of finding Exact Size Cut-Sets in Bipartite Graphs
I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
10
votes
1answer
338 views
What is the complexity of this game?
This is a generalization of my previous question.
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 ...
5
votes
1answer
106 views
Counting/Enumerating Minimal Edge Covers
A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it.
Questions
Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
10
votes
1answer
161 views
Linear circuit complexity classes
The class $\textrm{NC}^i$ is the class functions computable by circuits families of bounded fan-in, $n^{O(1)}$ size and $O(\log^i(n))$ depth.
The $\textrm{NC}$-hierarchy is the union of those classes....
3
votes
2answers
396 views
Is there a non-deterministic version of the complexity class PP?
From a quick skim of the literature (and complexity zoo), there doesn't seem to be a non-deterministic version of PP. Is there a reason for this (e.g. PP=non-deterministic PP?)
Edit: Perhaps I ...
10
votes
1answer
229 views
Is algorithmic information theory still evolving?
I am currently looking for a subject for a thesis and encountered the field of algorithmic information theory.
The field seems very interesting for me, but it seems everything is the field was done ...
8
votes
0answers
112 views
Complexity of fractional SAT
Let $(a, k)$-SAT be $k$-SAT with the promise that if there is there is a satisfying assignment, then there is such an assignment that satisfies at least $a$ literals of every clause. Can 3-SAT with $...
9
votes
0answers
146 views
Relatively low ambitious frontiers
What are some of the current "relatively" low ambitious frontiers for MA/PhD thesis in complexity theory class separations/containment or quantum computing?
For example: In the draft version of Arora ...
5
votes
1answer
177 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}/\...
7
votes
0answers
170 views
How many different proofs are there of parity is not in AC0?
The theorem that Parity is not in $\mathsf{AC}^0$ is one of the gemstones of complexity theory. I wonder how many different proofs there are of this result? What constitutes "different" is also a part ...
8
votes
0answers
221 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
0answers
34 views
How should a reduction to the Cardinality Constrained Quadratic Knapsack Problem work?
in Polyhedral Study of the Cardinality Constrained Knapsack Problem the authors prove that the Cardinality Constrained Knapsack Problem is NP-Hard by reducing PARTITION to it.
Besides, it's easy to ...
7
votes
1answer
195 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?
6
votes
0answers
92 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 ...
3
votes
1answer
97 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 ...
7
votes
0answers
52 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,...
0
votes
0answers
61 views
Proving that a problem is coNP-complete [duplicate]
Let $\le^{p}_{T}$ be the Turing or the Cook reduction and $\le_{m}^{p}$ the Karp-Levin reduction.
I know that to prove a problem $P_1$ is coNP-complete I just need to show that $P_1 \in coNP$ and ...
5
votes
0answers
81 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 ...
