24
votes
Accepted
Is there an NP-complete language that contains precisely half of the n-bit instances?
I asked this question a few years ago and Boaz Barak positively answered it.
The statement is equivalent to the existence of an NP-complete language $L$ where $|L_n|$ is polynomial-time computable.
...
- 1,801
22
votes
Accepted
Implications of proving NP=RP on complexity theory
Prelude: the below is just one consequence of $\mathsf{RP}=\mathsf{NP}$ and probably not the most important, e.g. compared to collapse of the polynomial hierarchy. There was a great and more ...
- 7,595
20
votes
Accepted
Intersection of languages in NP
Just an extended comment to better explain ARi's comment (I was writing it while I saw it).
It is sufficient to use a "large gap" approach similar to the one used in Lardner's theorem; for example:
$...
- 22.6k
19
votes
Accepted
Euclidean TSP in NP and square root complexity
Q1. This is a notorious open problem. It is known to be in the fourth level of the counting hierarchy, due to [ABKM]. Not known to be in NP. The problem is not really in computing square roots as ...
- 18.1k
17
votes
Accepted
Algorithm whose running time depends on P vs. NP
If you assume that $P=^?NP$ is provable in PA (or ZFC), a trivial example is the following:
...
- 22.6k
16
votes
Accepted
(How) Could we discover/analyze NP problems in the absence of the Turing model of computation?
You may wish to look at cost semantics for functional languages. These are various computational complexity measures for functional languages that do not pass through any kind of Turing machine, RAM ...
- 28.3k
15
votes
Accepted
How much would a SAT oracle help speeding up polynomial time algorithms?
Actually, acceptance of nondeterministic Turing machines in time $t$ is $O(t \log t)$-time reducible to SAT (the construction is via oblivious simulation, see Arora-Barak), so typically any time a ...
- 2,295
14
votes
Questions regarding SETH
The subtlety comes in where we introduce the notion of "harder". The reduction showing that SAT can be reduced to Hamiltonian Cycle shows that the latter is "harder" up to polynomial factors. In doing ...
- 1,642
14
votes
(How) Could we discover/analyze NP problems in the absence of the Turing model of computation?
At the request of Andrej and PhD, I am turning my comment into an answer, with apologies for self-advertising.
I recently wrote a paper in which I look at how to prove the Cook-Levin theorem ($\...
- 5,273
14
votes
Accepted
When was co-NP introduced for the first time?
Albert R. Meyer and Larry J. Stockmeyer introduced the polynomial hierarchy in 1972 with their paper "the equivalence problem for regular expressions with squaring requires exponential space"...
- 5,840
12
votes
Accepted
Are There Highly Symmetric NP- or P-complete Languages?
For NP, this seems hard to construct. In particular, if you can also sample (nearly) uniform elements from your group - which is true for many natural ways of constructing groups - then if an NP-...
- 36.9k
11
votes
Accepted
Is $\sf{P^{NP \cap coNP}} = \sf{NP \cap coNP}$?
First, this result is listed in the complexity zoo: https://complexityzoo.uwaterloo.ca/Complexity_Zoo:N#npiconp. Alternatively, it's possible to prove without much trouble (which I do below).
We want ...
- 2,758
11
votes
"Berman-Hartmanis Conjecture Separates NP From All Super-Poly. DTIME Classes" -- Worthy of arXiv.org?
I'm glad you are interested in complexity but there are some issues in your paper. Your techniques relativize and there is an oracle relative to which the Berman-Hartmanis conjecture is true and NP = ...
- 8,616
9
votes
Accepted
Enumerating finite set of words with Hamming distance $1$
This problem is NP-complete.
The class of graphs in the question is equivalent to the cubical graphs *1, but this class contains grid graphs.
Because the Hamiltonian path problem in grid graphs is NP-...
- 206
8
votes
Is there an NP-complete language that contains precisely half of the n-bit instances?
Here's a suggestion of why it might be difficult to come up with an example of such, though I agree with Kaveh's comment that it would be surprising if it didn't exist. [Not an answer, but too long ...
- 36.9k
8
votes
Accepted
Is there any relationship of hardness between the two problems?
No, you cannot infer hardness of P1. (And your question looks suspiciously close to homework.) Consider the special case where
$D$ is an undirected graph $G=(V,E)$
$x$ is a subset $E_x\subseteq E$
$...
- 5,772
8
votes
Natural NP-complete problems with high density?
Comment => Answer.
In this paper, Posa shows that for some constant $c$, a graph chosen from the Erdos-Renyi random graph distribution $G(n, c \log n / n)$ has a Hamiltonian cycle with probability ...
- 1,642
8
votes
Accepted
Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis
First of all, Mahaney's Theorem says that merely assuming $\mathsf{P} \neq \mathsf{NP}$, there are no sparse $\mathsf{NP}$-complete sets. (Historically, Mahaney was motivated to study this precisely ...
- 36.9k
8
votes
Accepted
Does p-isomorphism preserve phase transition?
Yes, but I'm not sure it means much. Yes in a trivial way: suppose $\varphi$ is an isomorphism between two $\mathsf{NP}$-complete languages $L_1, L_2$, and $L_1$ exhibits a phase transition with ...
- 36.9k
8
votes
Graph theoretic restriction to Proofs in Proof Complexity Theory
Müller and Szeider study Resolution proofs where the proof DAG has bounded tree-width or bounded path-width (for suitable extensions of these graph complexity measures to directed graphs.)
They show ...
- 4,621
8
votes
Accepted
List of NP-Complete graph problems/ properties?
A general list of NP-complete problems can be found in Garey & Johnson's book "Computers and Intractability". It contains an appendix that lists roughly 300 NP-complete problems, and despite its ...
- 918
7
votes
How much would a SAT oracle help speeding up polynomial time algorithms?
More generally, if we can pick any problem in NP−P, and use an oracle for it, then which of the problems in P could see a speed-up?
This question gets more directly at representation and time ...
- 7,595
7
votes
Problems in NP with non-trivial certificate
I feel like problems $P\in\mathsf{NP}\cap\mathsf{coNP}$ are good examples for your question.
Typically, for $P\not\in\mathsf{P}$, at least one of the witnesses is non-trivial.
For example, the closest ...
- 918
6
votes
Questions regarding SETH
Cygan, Kratsch and Nederlof give a $(2+\sqrt{2})^{\texttt{pw}} n^{O(1)}$ algorithm for hamiltonicity on graphs with $n$ vertices and pathwidth $\texttt{pw}$ (assuming you are given the pathwidth ...
- 14.3k
6
votes
Graph theoretic restriction to Proofs in Proof Complexity Theory
For strong enough proof systems the graph representation of a proof in the system seems less consequential, since (as Joshua Grochow already commented), DAG-like and tree-like Frege proofs are ...
- 1,899
6
votes
Accepted
Reduction of graph chromatic number to hypergraph 2-colorability
As the other answer points out, the reduction in the original paper seems to have a bug: $H$ will not be two-colorable unless $G$ is bipartite. I couldn't quite see how to prove the reduction in the ...
- 18.1k
6
votes
Implications of proving NP=RP on complexity theory
A simple answer is that we're "pretty sure" that $\mathsf{P} \neq \mathsf{NP}$, and we're "pretty sure" that $\mathsf{P} = \mathsf{RP}$, so we're "pretty sure" that $\...
- 4,011
6
votes
Accepted
Conversion between NP certificates
For general $L\in\mathrm{NP}$, this is equivalent to $\mathrm{FP=TFNP}$, hence likely false:
On the one hand, if $V$ and $V'$ are verifiers of $L$, then “given $x$ and $u$ such that $V(x,u)$, find $u'$...
- 16.8k
5
votes
Is there a non-deterministic linear time algorithm for CNF-SAT?
This is only an extended comment.
A few times ago I asked (myself :-) how fast a multitape NTM that accepts a (reasonably encoded) NP-complete language can be. I came up with this idea:
3-SAT ...
- 22.6k
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