P versus NP and other resource-bounded computation.

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4
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1answer
44 views

Can differential equations be classed into their own complexity classes?

Problems have been, as a whole, classified, thanks to Computational Complexity. But, in differential equations, is it possible to classify differential equations depending on their computational ...
5
votes
0answers
151 views

Partition of a set of integers into subsets with prescribed sums

I saw this problem: A non increasing sequence of positive integers $m_1,m_2,..., m_k$ is said to be n-realizable if the set $I_n=\{1,2,..., n\}$ can be partitioned into $k$ mutually disjoint subsets ...
0
votes
0answers
79 views

Proof of Non Deterministic Space Hierarchy Theorem [on hold]

I am trying to prove the Non Deterministc Space Hierarchy Theorem, which says: If $f$ and $g$ are two functions such that $f=o(g)$ where $g$ is fully space constructible and $g(n) \ge \text {log }n$, ...
24
votes
6answers
2k views

Problems with big open complexity gaps

This question is about problems for which there is a big open complexity gap between known lower bound and upper bound, but not because of open problems on complexity classes themselves. To be more ...
-2
votes
1answer
101 views

Why can't Horn-SAT be solved in Log-space? [closed]

A simple algorithm for Horn-SAT (in CNF) is the following: Given: A Horn formula $\phi$ in CNF. Find a unit clause (a clause with one literal) $C_i$. $~$Set the variable $x_j$ appearing in $C_i$ to ...
0
votes
1answer
62 views

Many-one reduction from inequality problem to equality problem

Let the k-inequality-MIS problem be the decision problem whether an arbitrary graph $G=(V, E)$ contains a maximal independent set of at least size $k$, that is the corresponding language is: ...
29
votes
2answers
508 views

Is there an oracle such that SAT is not infinitely often in sub-exponential time?

Define $io$-$SUBEXP$ to be the class of languages $L$ such that there is a language $L' \in \cap_{\varepsilon > 0} TIME(2^{n^{\varepsilon}})$ and for infinitely many $n$, $L$ and $L'$ agree on all ...
-2
votes
0answers
55 views

$FNP \subset FPSPACE$? [closed]

it is clear, that $NP \subseteq PSPACE$ holds and that it is unknown if the strict inclusion holds. how is it if one looks at the corresponding functional complexity classes? Does $FNP \subset ...
4
votes
1answer
153 views

possible bridge between group growth theory and complexity theory?

RJ Lipton conjectures a link between group growth theory and complexity theory. Group growth theory has undergone rapid advance in the last decade and has many surface similarities/ parallels with ...
6
votes
1answer
110 views

Nondeterministic communication complexity of set disjointness?

In the two-party setting, bounds of $\Theta(n)$ bits are known for deterministic and bounded-error randomized protocols for $\text{DISJ}_n$. (Here $\text{DISJ}_n$ is the $n$-element set disjointness ...
-2
votes
1answer
107 views

Deciding CL-IS on graph efficiently

Given an arbitrary graph $G$, could there be a polynomial time algorithm to tell if it has a larger size clique $(\omega(G))$ or larger independence number$(\alpha(G))$?
4
votes
1answer
120 views

Introductions to steganography from an information-theoretic standpoint

Can I get some introductory references for steganography from an information-theoretic standpoint? I recently listened to a talk on it, and the speaker said that he knew of no good introductions to ...
1
vote
1answer
112 views

Comparison between the maximum clique and maximum biclique problem

It seems to be commonly believed that the maximum biclique problem (on a bipartite graph) is more difficult than the original maximum clique problem. Is there a formal proof for this claim? (For ...
-1
votes
0answers
29 views

Volume complexities of multihead Turing Machines

I'm trying to prove that for every multihead Turing machine X, there is a multihead Turing machine y such that for any input string z, we have volume(X, z) = Θ(Y(z)) and volume(Y,z) = Θ(Y(z)). In ...
4
votes
0answers
86 views

An oracle relative to which EXP(NP) = BPP

Whether or not $\mathbf{BPP} = \mathbf{EXP}^{\mathbf{NP}}$ is an open problem, although we believe the former is strictly contained in the other. I guess, from the absence of the proof of the ...
18
votes
3answers
1k views

If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
0
votes
1answer
218 views

NP-complete problems with optimal approximation in poly-time

I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not ...
20
votes
5answers
561 views

Curious about computer-assisted NP-completeness proofs

In the paper "THE COMPLEXITY OF SATISFIABILITY PROBLEMS" by Thomas J. Schaefer, the author has mentioned that ...
0
votes
1answer
99 views

What is the Algorithm to find all the possible chordal graphs which can be formed by a given 'n' number of vertices

A chordal Graph is a connected graph which contains no chord-less cycle of size greater than three. They are also called as Triangulated graphs. All Paths are Chordal Graphs (No cycles). All Trees ...
9
votes
1answer
461 views

Is the complexity of this path problem known?

Instance: An undirected graph $G$ with two distinguished vertices $s\neq t$, and an integer $k\geq 0$. Question: Does there exist an $s-t$ path in $G$, such that the path intersects at most $k$ ...
4
votes
1answer
330 views

The complexity of a multi-objective shortest path problem

I have the following shortest path problem. Consider a directed graph with $n$ levels. Each level has $m$ nodes. Each node at level $i$ is connected to all nodes at level $i+1$. Let us also make a ...
14
votes
1answer
356 views

Integer linear programming in logarithmic number of variables

I read that integer linear programming is solvable in polynominal time if the number $n$ of variables is fixed, i.e. $n \in O(1)$. If the number of variables grows logarithmically, i.e. $n \in ...
10
votes
2answers
238 views

Which graph problems are $W[1]$-Hard on directed(/weighted) graphs but FPT on undirected(/unweighted) graphs?

Following the equivalent questions regarding NP-Completeness (see the weight question and the directed question), I was wondering how parameterized problems are affected by these attributes. ...
5
votes
1answer
66 views

Quanitifier Free Presburger Arithmetic: Upper bound on solution size?

DISCLAIMER: I had originally posted this to CS.SE, but I've deleted it and moved it here, since it received little attention, and I think it is a research level question. According to this paper, if ...
3
votes
1answer
119 views

The relationship between completeness and strength of reductions

Ladner theorem can be stated as: $P \ne NP$ if and only if there exists an incomplete set in $NP-P$. Here an incomplete set is a set that is not complete for $NP$ under many-one polynomial time ...
5
votes
0answers
156 views

Evidence that Graph Isomorphism problem is not $NP$-complete

Graph isomorphism problem is one of the longest standing problems that resisted classification into $P$ or $NP$-complete problems. We have evidences that it can not be $NP$-complete. Firstly, Graph ...
2
votes
0answers
59 views

$\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ problems

All the problems which are either $\mathsf{APX}-\mathsf{Hard}$ and $\mathsf{MaxSNP}-\mathsf{Hard}$ cannot admit a $\mathsf{PTAS}$, unless $P=NP$. I would like to know whether $\mathsf{MaxSNP}$ ...
3
votes
1answer
113 views

What is the relationship between $\mathsf{APX}$ and $\mathsf{MaxSNP}$ classes?

My understanding of these classes is a really fuzzy. The more I am trying to read the more I am getting confused. Can anyone help me understand the relationship between these classes. More precisely, ...
5
votes
0answers
106 views

Implications of a deterministic polytime prime-finding algorithm

I'm wondering what are the current known uses/implications of a polynomial-time algorithm for the following problem: Given $n$ in binary, output a prime $p > n$. I'm both curious about ...
11
votes
2answers
213 views

What is the complexity of the equivalence problem for read-once decision trees?

A read-once decision tree is defined as follows: $True$ and $False$ are read-once decision trees. If $A$ and $B$ are read-once decision trees and $x$ is a variable not occurring in $A$ and $B$, ...
4
votes
1answer
106 views

Deciding whether the sum of independent random variables exceeds a threshold a majority of the time, PP-hard?

Say I have $n$ independent Bernoulli random variables, with parameters $p_1,\ldots,p_n$. Say, also, that I wish to decide whether their sum exceeds some given threshold $t$ with probability at least ...
1
vote
0answers
37 views

Computational complexity of Initial Value Problems of ODEs

Are there known results on computational complexity of initial value problems of ODEs? As my question may be somewhat vague, I want to mention that I'm mainly interested for results on the ...
6
votes
1answer
141 views

Integer factorization using polynomial whose roots are prime factors

Let $n$ be a square-free positive integer, let $n=p_{1}p_{2}\ldots p_{k}$ be the prime factorization of $n$ into $k$ distinct primes $p_{i}$. For such $n$, define ...
1
vote
0answers
114 views

Consequences of the existence of the following algorithm: does it imply any complexity class separation / collapse?

Let $G$ be a $3$-regular graph. Let $O$ be the number of vertex covers of $G$ having odd cardinality, and let $E$ be the number of vertex covers of $G$ having even cardinality. Let $\Delta = O - E$. ...
4
votes
1answer
155 views

Graph theoretic restriction to Proofs in Proof Complexity Theory

Proof complexity is a most basic area of computational complexity theory. An ultimate purpose of this area is to prove $NP\neq coNP$, that is, any prover cannot give a proof of unsatisfiability of ...
8
votes
2answers
687 views

Is square removal easier than factoring?

It seems to me that the square removal task can be reduced to the factoring task, but that there is no way to reduce factoring to square removal. Is there a way to make this "feeling" more precise, ...
0
votes
1answer
41 views

polytime transformation from a graph to a set of binary strings

$d_H$ denotes the Hamming distance between two binary strings of the same size. The problem is stated as follows. Given any undirected graph $(V, A)$, does there always exist a one-to-one ...
3
votes
3answers
221 views

Minimum vertex cover on k-regular graphs, for fixed k>2 NP-hard proof?

Good Evening! I am aware that the minimum (cardinality) vertex cover problem on cubic graphs ($3$-regular) graphs is $NP$-hard. Say positive integer $k>2$ is fixed. Has there been any ...
5
votes
1answer
132 views

Downward self-reducibility of factorization

Is integer factorization downward self-reducible? Is anything known about this?
2
votes
1answer
46 views

Promise Variant of Set-Packing

An instance of the SET-PACKING problem is given by a list of sets $\mathcal{S} = \{S_1,\dots,S_m\} \subseteq 2^U$. It is a ``yes'' instance iff there exists some subset $\mathcal S'$ of $\mathcal S$ ...
2
votes
1answer
80 views

Extractors in Practice: How to Determine the Min-Entropy in the Source Distribution

One of the main parameters in the construction of extractors is $k$, the min-entropy of the source distribution. In practice, suppose we want to extract randomness from a given source $S$. How do we ...
10
votes
0answers
207 views

Computing $\operatorname{MAJ}_n$ by $\operatorname{MAJ}_m$ in depth 2

Can the majority of $n$ bits be computed by a depth 2 formula all of whose gates compute the majority of $m$ bits where $m=O(n^c)$ for a constant $c<1$? Such a formula contains $m+1$ gates and ...
0
votes
1answer
86 views

How does one extend local checkability to quantum complexity classes?

How does one extend local checkability to quantum complexity classes like BQP? Or are quantum algortihms inherently holistic?
0
votes
1answer
123 views

Idea for a white-box PIT deterministic algorithm in polynomial time

Just a warning, I am an amateur and this algorithm probably doesn't work. A high level description of the algorithm: Be $f(x_1,...,x_n)$ a polynomial, $n$ the number of variables and $d$ is the ...
5
votes
0answers
81 views

Is the finite inverse semigroup isomorphism problem GI-complete?

Is the finite inverse semigroup isomorphism problem GI-complete? Here the finite inverse semigroups are assumed to be given by their multiplication tables.
4
votes
2answers
275 views

Is there a polynomial time algorithm for creating a set of vectors in general position?

It seems to be possible to create a set of vectors in $\mathbb{R}^n$ such that any subset of $n$ vectors forms a basis. For example, in $\mathbb{R}^3$, here is such a set with 21 vectors: $$ \left\{ ...
5
votes
2answers
289 views

Complexity class of sensitivity

Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a Boolean function. Let $[n]=\{1,2,\dots,n\}$. If $i\in[n]$, let $\Bbb 1_i$ be length $n$ vector with all $0$s except $1$ at $i$th position. If $B\subseteq ...
21
votes
1answer
469 views

Complexity of n-queens-completion?

The classical $n$-queens problems asks, given a positive integer $n$, whether there is an array $Q[1..n]$ of integers satisfying the following conditions: $1\le Q[i] \le n$ for all $i$ $Q[i] \ne ...
3
votes
0answers
322 views

$NP$-complete problems on cubic Hamiltonian graphs

The class of cubic Hamiltonian graphs is well studied class. I came across the fact that independent set problem is $NP$-complete when restricting input to cubic Hamiltonian graphs. I am interested in ...
10
votes
1answer
187 views

Hartmanis-Stearns conjecture and the computable transcendental numbers

In the 1965 article "On the computational complexity of algorithms" by Hartmanis and Stearns, the authors conjecture that if a real-time Turing Machine computes the real number $r$ in, for example, ...