Computer Science Stack Exchange
Computer Science Stack Exchange

Questions tagged [np]

Questions about decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input.

NP is the class of decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input. Equivalently, these are decision problems where every "yes" instance has a proof of this fact, which has length polynomial in the size of the instance and can be verified deterministically in polynomial time.

One of the major open problems in computer science is whether P=NP. That is, whether deterministic polynomial time Turing machines have the same power as nondeterministic ones. Or, informally, whether being able to easily compute the answer to a problem (P) is the same as being able to easily verify an answer, in the sense given above. The Clay Mathematics Institute has offered a million-dollar prize for the resolution of this question.

For more information, see our reference question.

792 questions
67
votes
2 answers

Are there subexponential-time algorithms for NP-complete problems?

Are there NP-complete problems which have proven subexponential-time algorithms? I am asking for the general case inputs, I am not talking about tractable special cases here. By sub-exponential, I mean an order of growth above polynomials, but…
ksb
  • 801
  • 1
  • 7
  • 5
35
votes
2 answers

Is there a task that is solvable in polynomial time but not verifiable in polynomial time?

A colleague of mine and I have just hit some notes of one of our professors. The notes state that there are tasks that are possible to solve in polynomial time (are in the class of PF) but that are NOT verifiable in polynomial time (are NOT in the…
Drozi
  • 351
  • 3
  • 3
27
votes
3 answers

NP-complete problems not "obviously" in NP

It occurred to many that in all the $\textbf{NP}$-completeness proofs I've read (that I can remember), it's always trivial to show that a problem is in $\textbf{NP}$, and showing that it is $\textbf{NP}$-hard is the... hard part. What…
gardenhead
  • 2,240
  • 14
  • 19
26
votes
13 answers

False proofs that look correct

I remember seeing a list of False Proofs when I was taking Discrete Maths and I found it to be very interesting and also helpful. So, if anyone knows some common proof mistakes students make or some cool sneaky way to trick people in a proof send…
proof-of-correctness
  • 572
  • 4
  • 9
24
votes
3 answers

Why are NP-complete problems so different in terms of their approximation?

I'd like to begin the question by saying I'm a programmer, and I don't have a lot of background in complexity theory. One thing that I've noticed is that while many problems are NP-complete, when extended to optimization problems, some are far more…
GregRos
  • 525
  • 2
  • 9
23
votes
2 answers

How is the traveling salesman problem verifiable in polynomial time?

So I understand the idea that the decision problem is defined as Is there a path P such that the cost is lower than C? and you can easily check this is true by verifying a path you receive. However, what if there is no path that fits this…
wjmccann
  • 579
  • 5
  • 15
23
votes
5 answers

Assuming P = NP, how would one solve the graph coloring problem in polynomial time?

Assuming we have $\sf P = NP$, how would I show how to solve the graph coloring problem in polynomial time? Given a graph $G = (V,E)$, find a valid coloring $\chi(G) : V \to \{1,2,\cdots,k\}$ for some $k$ satisfying the property that $(u,v) \in E$…
donkey
  • 352
  • 2
  • 9
19
votes
2 answers

Why are computability problems always written in full caps?

Maybe this is an odd question. It has always bugged me that computability problems are written in all caps, and in such an "awkward" way. SAT, 3-SAT, COLORING, 3-COLORING, PARTITION, CLIQUE, VERTEX COVER. Etc. They're not proper titles, just single…
user162598
17
votes
1 answer

What does the 2 in a 2-approximation algorithm mean?

Does the 2 in a 2-approximation algorithm mean the solution is within 2*OPT or OPT/2?
Hrishikesh
  • 171
  • 1
  • 1
  • 3
17
votes
4 answers

Why is Integer Linear Programming in NP?

The decision version of the problem Integar Linear Programming is the following: Input: two matrices $A\in \mathcal{M}_n(\mathbb{Z})$ and $B\in \mathcal{M}_{n,1}(\mathbb{Z})$. Question: is there a matrix $X\in \mathcal{M}_{n,1}(\mathbb{Z})$ such…
Nathaniel
  • 18,309
  • 2
  • 30
  • 58
16
votes
3 answers

Is there a complexity viewpoint of Galois' theorem?

Galois's theorem effectively says that one cannot express the roots of a polynomial of degree >= 5 using rational functions of coefficients and radicals - can't this be read to be saying that given a polynomial there is no deterministic algorithm…
user6818
  • 1,165
  • 8
  • 13
16
votes
2 answers

What is so fundamental about polynomial functions that they are used to demarcate the Hardness boundary in NP complexity classes?

This question has been bugging me ever since I first came across the concept of NP, NP-Complete, and NP-Hard a few years back: what is so fundamental about the polynomial functions that they are used to demarcate the boundary between what is "hard"…
shivams
  • 261
  • 1
  • 5
15
votes
4 answers

Proof Complexity of a Proof or Disproof of P = NP

Has there been any research on the proof complexity of a resolution to the P=NP problem? If not, given the lack of progress on the problem, would it be unreasonable to conjecture that any proof that resolves the P=NP problem will require a…
Tony Johnson
  • 153
  • 1
  • 6
15
votes
4 answers

Can any finite problem be in NP-Complete?

My lecturer made the statement Any finite problem cannot be NP-Complete He was talking about Sudoku's at the time saying something along the lines that for a 8x8 Sudoku there is a finite set of solutions but I can't remember exactly what he said.…
TheRapture87
  • 271
  • 2
  • 4
14
votes
1 answer

Is determining if there is a prime in an interval known to be in P or NP-complete?

I saw from this post on stackoverflow that there are some relatively fast algorithms for sieving an interval of numbers to see if there is a prime in that interval. However, does this mean that the overall decision problem of: (Does there exists a…
Ari
  • 1,661
  • 10
  • 23
1
2 3
…
52 53