P versus NP and other resource-bounded computation.

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Maximum number of configurations on a TM that decides the language $A_\text{NFA}$ [migrated]

Consider a Turing Machine $M$ that decides the following language: $$A_{\text{NFA}} = \{ \langle N,w \rangle | N\text{ is an NFA and }N\text{ accepts }w \}.$$ Based on its input size, if $M$ wants to ...
7
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184 views

Are there any implementations of Cook-Levin?

Does anyone know if there is an implementation of Cook-Levin? A program that gets an input like the following: $M$: the code of a machine model/simple program in a programming language like C, ...
9
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0answers
151 views

Does Kannan's theorem imply that NEXPTIME^NP ⊄ P/poly?

I was reading a paper of Buhrman and Homer “Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy”. On the bottom of page 2 they remark that the results of Kannan imply that ...
7
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1answer
146 views

Consequences of nondeterminism speeding up deterministic computation

If $\mathsf{NP}$ contains a class of superpolynomial time problems, i.e. for some function $t \in n^{\omega(1)}$, $\mathsf{DTIME}(t) \subseteq \mathsf{NP}$, then if follows from the ...
6
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3answers
335 views

Exponential gap on neural network layers

I read it here that there are function families which need $\mathcal{O}(2^n)$ nodes on neural network with at most $d - 1$ layers to represent the function while need only $\mathcal{O}(n)$ if the ...
9
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1answer
223 views

Are there non-constructive proofs of existence of “small” Turing machines / NFAs?

After reading a related question, about non-constructive existence proofs of algorithms, I was wondering if there are methods of showing existence of "small" (say, state-wise) computation machines ...
10
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122 views

Approximating $\textrm{AC}^{0}$ by sparse polynomials

Let $f$ be a Boolean function from $\{0,1\}^{n}$ to $\{0,1\}$. We say that $f$ is randomly approximated with error probability $\epsilon$ by a family of polynomials $P$ if \begin{equation} \forall ...
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0answers
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Why use languages in Complexity theory [migrated]

I'm just starting to get into the theory of computation, which studies what can be computed, how quickly, using how much memory and with which computational model. I have a pretty basic question, but ...
25
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1answer
425 views

Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
12
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2answers
207 views

Provable gaps between decision tree complexity and “true” complexity

The title is a little misleading: but hopefully the question isn't: Grønlund and Pettie's new result showing that 3SUM has only $O(n^{3/2})$ decision tree complexity got me wondering: Is there a ...
19
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3answers
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Has there been any research on $k$-SAT above the satisfiability threshold?

A well known characteristic of $k$-SAT instances is the ratio of the number of clauses $m$ over the number of variables $n$, i.e., the quotient $\rho = m/n$. For every $k$, there is a threshold value ...
19
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1answer
474 views

Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
3
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0answers
93 views

For which values of $k$ is minimum length undirected $k$-disjoint-paths in $\mathcal{P}$?

In a related question, Saeed and Super8 have mentioned the Robertson-Seymour theory which enables us to find $k$ disjoint paths between pairs of vertices $\{s_i,t_i\}_{i=1}^k$ in poly time for fixed ...
5
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0answers
102 views

Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
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2answers
455 views

Comparing Two Algorithms for 3SUM problem over Integers

The paper "Subquadratic Algorithms for 3SUM", by Ilya Baran, Erik D. Demaine, Mihai Patrascu has the following complexity for the 3SUM problem: given a list $L$ of $n$ integers if there are $x,y,z ...
24
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1answer
271 views

Can graph isomorphism be decided with square root bounded nondeterminism?

Bounded nondeterminism associates a function $g(n)$ with a class $C$ of languages accepted by resource-bounded deterministic Turing machines, to form a new class $g$-$C$. This class consists of those ...
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2answers
170 views

On the status of learnability inside $\mathsf{TC}^0$

I'm trying to understand the complexity of functions expressible via threshold gates and this led me to $\mathsf{TC}^0$. In particular, I'm interested what's currently known about learning inside ...
4
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70 views

Is minimum weight simple cycles through specified vertics fixed parameter tractable?

The problem formulation is as follows: Input: Undirected graph $G=(V,E)$, a set of vertices $S\subseteq |V|$, a weight function $w:E\to \mathbb{R}$ and a threshold $T\in \mathbb{R}$. ...
12
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5answers
631 views

Should reductions make us more or less optimistic for the tractability of a problem?

It seems to me that most complexity theorists generally believe the following philosophical rule: If we can't figure out an efficient algorithm for problem $A$, and we can reduce problem $A$ to ...
3
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1answer
125 views

$NP$ set as a function of combinatorial space

Inspired by this answer given by Noam, which I think implies (if I understood correctly his answer ) that a set $A \in NP$ if and only if there is polynomial-time computable function $f$ from a ...
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136 views

Is SAT in $c^{\sqrt{n}}$ time with preprocessing worthwhile?

We may be able to solve SAT in $c^{\sqrt{n}}$ for $n$ variables and a constant $c$. Now we can suppose that we require a certain amount of preprocessing to get this result. For example, in this ...
14
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1answer
305 views

Does PSPACE-completeness imply approximation hardness?

It is mentioned in a comment in another cstheorySE post that PSPACE-completeness imply APX-hardness. Can anyone please explain/share a reference for it? Is this "tight"? (i.e., are there ...
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1answer
68 views

PAC algorithms for APX-Hard problems

Do there exist polynomial time algorithms that admit Probably Approximately Correct (PAC) bounds for APX-Hard problems? That is, does there exist a problem $P$ that is APX-Hard, such that for every ...
25
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1answer
561 views

Computational complexity of pi

Let $L = \{ n : \text{the }n^{th}\text{ binary digit of }\pi\text{ is }1 \}$ (where $n$ is thought of as encoded in binary). Then what can we say about the computational complexity of $L$? It's ...
7
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2answers
138 views

Planted Clique in G(n,p), varying p

In the planted clique problem, one must recover a $k$-clique planted in an Erdos-Renyi random graph $G(n,p)$. This has mostly been looked at for $p=\frac{1}{2}$, in which case it is known to be ...
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1answer
142 views

Complexity of a parametrized min-cost flow problem

Consider the following min-cost flow variant: Input: a positively-weighted complete bipartite graph $G = (S, T, c)$ and extra vertices $s, t$ edges from $s$ to $S$ and from $T$ to $t$. lower ...
9
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1answer
239 views

APX Hardness implies no QPTAS?

So, a quick search on the web led me to believe that "APXHardness implies that no QPTAS exist for a problem unless [some complexity class] is included in some [other complexity class]" and it is well ...
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50 views

Different hardness proofs w.r.t different classes

Consider a language $L$ which is hard for some class $C$ (e.g. PSPACE-hard). Trivially, $L$ is also $D$-hard for every class $D\subseteq C$ under the same type of reduction (e.g. NP-hard). Is there a ...
5
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2answers
236 views

Baker Gill Solovay $P^B \ne NP^B$ relativization, what class is $B$ in?

A recent question asks whether relativization is well-defined. This question wonders how to put one use of it on firmer ground. In the BGS 1975 proof that there exists a language $B$ such that ...
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87 views

NP algorithm for an optimization problem

Consider the following optimization problem INSTANCE: vectors $\vec{a}, \vec{b}, \vec{c}$, matrices $A, B$, threshold $\theta$. PROBLEM: Let $f={\displaystyle \max_{\vec{x}, \vec{y}}} ...
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1answer
102 views

Parallel (NC) replacements for Gaussian elimination?

Suppose one has an matrix $A$ with $c$ columns and $r$ rows with entries in the binary field $GF(2)$. One wants to determine its rank, and a basis for its null-space. These can be computed ...
11
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1answer
243 views

$\mathsf{NP} \cap \mathsf{coNP}$ as oracle

Does $\mathsf{NP^{NP \,\cap\, coNP}=NP}$ hold? Clearly $\mathsf{NP^{NP}\neq NP}$, but it seems to me that $\mathsf{NP\cap coNP}$ is "deterministic" which makes me believe this is true. Is there a ...
4
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221 views

Notion of associativity for ternary relations?

Consider the problem $\def\GEN{\mathrm{GEN}}\GEN$: given a ternary relation $R \subseteq X^3$ and a subset $S \subseteq X$, is the closure of $S$ with respect to $R$ equal to the whole set $X$? By ...
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1answer
88 views

Polynomial-time distinguishability threshold of planted clique

I have a basic question regarding the best known polynomial-time "distinguishing advantage" for the planted clique problem. By this, I'm referring to the problem of distinguishing the distribution ...
6
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2answers
222 views

Is relativization well-defined?

According to BGS theorem [1], there is an oracle $A$ such that $P^A\neq NP^A$. If the relativization operation $B\mapsto B^A$ was a well-defined function, one would expect that from $B^A\neq C^A$ ...
4
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1answer
99 views

Subset Sum Variant still NP hard?

I've read that this SE is for research-level CS questions; this is certainly research level, just not my field of research, so it is hard for me to assess how difficult the question really is. So in ...
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1answer
204 views

NP completeness over reals

I am studying the BSS model of computation recently (cf. for instance Complexity and Real Computation; Blum, Cucker, Shub, Smale.) For reals $R$, it is shown that, given a system of polynomials ...
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0answers
165 views

Does $P\neq NP$ imply any larger separation?

I've asked a similar question in cs.se, but didn't get a satisfying answer. Assuming $P\neq NP$, what can we say about the runtime of any algorithm for an $NP$-complete problem? Obviously, it means ...
11
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2answers
306 views

Is NP in $DTIME(n^{poly\log n})$?

Is NP in $DTIME(n^{poly\log n})$?
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1answer
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3
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1answer
95 views

How do you compute the fixed point of a best-response function efficiently?

I have a polynomial time best-response function that has the same properties as a game-theory game (convexity, compactness, set-valued). I don't know that much topology, but my understanding is that ...
16
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2answers
395 views

Problems with efficient solution except for a small fraction of inputs

The halting problem for Turing machines is perhaps the canonical undecidable set. Nevertheless, we prove that there is an algorithm deciding almost all instances of it. The halting problem is ...
5
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3answers
334 views

If SAT is in PCP, for some constant q, then P = NP

I have seen this statement before, but I haven't really seen a proof of it: If $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, for some constant $q$, then $P = NP$. Now, if $SAT\in PCP_{1,2^{−q}}[\log(n),q]$, ...
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2answers
125 views

Set packing with maximum coverage objective

We are given a universe $\mathcal{U}=\{e_1,..,e_n\}$ and a set of subsets $\mathcal{S}=\{s_1,s_2,...,s_m\}\subseteq 2^\mathcal{U}$. Set-Packing asks how many disjoint sets we can pack, and is defined ...
5
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1answer
161 views

Grover's search algorithm for 3 coloring

According to Arora & Barak (pdf), pg. 186, for a polynomial-time computable function $f: \{0,1\}^n \to \{0,1\}$ (represented as a circuit computing $f$), Grover's algorithm finds in ...
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1answer
120 views

Figuring EasyVer problems - problems whose witness can be verified in time independent on the instance size

In a related question I've defined a class of graph problems which are verifiable using a time related only on the size of the witness: $EasyVer=\{L\subset \mathcal{G}\times \mathbb{N}| $ a witness ...
4
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89 views

Low space computation and branching program

One of the most elemental result of relationship between boolean circuit size and polynomial uniform computation is Pippenger and Fishers simulation: $DTIME[T(n)]\subseteq SIZE[T(n)\log T(n)]$. I ...
4
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1answer
119 views

Is it possible to create an algorithm-aware optimizer?

I've recently implemented a physics system where each object has to interact with eachother. It consisted of, pretty much, the following algorithm: ...
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1answer
307 views

Best sources for communication complexity

What are some of the best sources (books and papers) to motivate and learn communication complexity on its own and in connection with its relation to computational complexity theory?
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128 views

What relations are there between a problem hardness and the hardness of verifying a witness?

I had some hard times trying to formulate the question, so I'll start with some examples: Suppose you are given a Dominating Set instance, $<G,k>$. Now suppose I give you a set of vertices $D$ ...