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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. ...
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 ...
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19 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: $...
19 votes

Graph theoretic restriction to Proofs in Proof Complexity Theory

The most natural restriction on the proof DAG is that it be a tree – that is, any "lemma" (intermediate conclusion) is not used more than once. This property is called being "tree-like". General ...
  • 14.2k
19 votes
Accepted

If BQP contains NP, does this mean that P=NP?

No, $\mathrm{NP}\subseteq\mathrm{BQP}$ is not known to imply $\mathrm P=\mathrm{NP}$. Even the stronger assumption $\mathrm{NP}\subseteq\mathrm{BPP}$ is not known to yield a deeper collapse than $\...
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 ...
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: ...
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 ...
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 ...
  • 2,305
15 votes

Integer linear programming in logarithmic number of variables

I can only give a partial answer to this question. A result by Lenstra (later improved by Kannan, and Frank and Tardos) states that ILP with $k$ variables can be solved in time $k^{O(k)}$ (times a ...
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 ...
  • 1,643
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 ($\...
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"...
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-...
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 ...
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 = ...
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-...
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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 ...
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 ...
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 ...
  • 1,643
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 $E_x\subseteq E$ $...
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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 ...
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 ...
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 ...
  • 778
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 ...
  • 7,140
7 votes
Accepted

Is there a simple game with asymmetric complexity?

Perhaps a fairly natural game is the following: Player 1 is placed in the middle of a maze and must reach the exit in order to win. Player 2 is in the same maze and must collect a set of "components"...
6 votes

Bounds on the size of the solution of a quadratic program

In Quadratic Programming is in NP, it is shown that a slight variant of this problem (QPL) is in NP. The question can be formulated as follows: does there exist a point $x \in \mathbb{R}^n$ such that $...
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 ...
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 ...
  • 14.2k
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 ...

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