44
votes
Is there a task that is solvable in polynomial time but not verifiable in polynomial time?
This is only possible if there are many admissible outputs for a given input. I.e., when the relation $R$ is not a function because it violates uniqueness.
For instance, consider this problem:
...
- 14.4k
42
votes
Accepted
Assuming P = NP, how would one solve the graph coloring problem in polynomial time?
There are two cases:
$P = NP$ non-constructively: this means we have derived a contradiction from the assumption that $P \neq NP$, and thus can conclude that $P = NP$ by the law of the excluded ...
- 29.6k
36
votes
Accepted
How is the traveling salesman problem verifiable in polynomial time?
NP is the class of problems where you can verify "yes" instances. No guarantee is given that you can verify "no" instances.
The class of problems where you can verify "no" instances in polynomial ...
- 81.4k
27
votes
Accepted
Can any NP-Complete Problem be solved using at most polynomial space (but while using exponential time?)
Generally speaking, the following is true for any algorithm:
Suppose $A$ is an algorithm that runs in $f(n)$ time. Then $A$ could not take more than $f(n)$ space, since writing $f(n)$ bits requires $...
- 1,659
26
votes
Why are computability problems always written in full caps?
After the Cook-Levin Theorem Richard Karp realized that the complexity of computational problems could be compared.
His paper was prepared in a type-writer font, and used underlining and all-caps for ...
- 29.6k
24
votes
Proof Complexity of a Proof or Disproof of P = NP
Proof complexity only makes sense when there is a sequence of statements depending on a parameter $n$. For example, the proposition $\mathsf{PHP}_n$ states (informally) that there is no bijection $[n+...
- 274k
23
votes
Can a subset of an NP-complete problem be in P?
Your question doesn't make sense:
The problem is NP-complete (proven) for all input data (without exception).
This is not a thing. NP-completeness is a property of entire sets, not of specific ...
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22
votes
Accepted
Evolving artificial neural networks for solving NP problems
No. This direction is unlikely to be useful, for two reasons:
Most computer scientists believe that P $\ne$ NP. Assuming P $\ne$ NP, this means there does not exist any polynomial-time algorithm to ...
D.W.♦
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20
votes
Accepted
Is determining if there is a prime in an interval known to be in P or NP-complete?
So your problem is as follows:
Input: integers $\ell,u$
Question: does there exist a prime in $[\ell,u]$?
As far as I know, it is not known whether that problem is in P or not.
Here's what I do know:
...
D.W.♦
- 154k
19
votes
Accepted
Is Post Correspondence Problem in NP?
The Post correspondence problem is undecidable, and in particular it is not in NP. The reason that your idea doesn't work is that the witness is not necessarily of polynomial size (in fact, you just ...
- 274k
18
votes
False proofs that look correct
One of my favourites is the "brothers paradox":
https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
I tell it as I learned it*, as follows:
in a village, each family has two children, elder ...
- 17k
17
votes
Accepted
Is detecting easy instances of NP-hard problems easy?
The problem isn't really well-posed. For any particular instance, there is a single solution, say $S$. Consequently, we can imagine an algorithm that has the answer $S$ hardcoded in: no matter what ...
D.W.♦
- 154k
17
votes
Accepted
Why rectangle packing is NP-hard but maybe not in NP?
In order for a language $L$ to be in NP, there needs to be a way to certify that instance $x$ belongs to $L$. This "way" is a polynomial size witness which can be verified in polynomial time....
- 274k
16
votes
Accepted
is FIND WORDS in P?
Your language is in P. Suppose that the matrix is $n\times n$ and that the words have total length $\ell$. Each word can start at at most $n^2$ positions and be written in $O(1)$ many orientations, ...
- 274k
16
votes
False proofs that look correct
Merge-sort can be done in linear time!
Indeed, the time complexity to sort a list or array of length $n$ verifies$^{(1)}$:
$$T(n) = T\left(\left\lfloor\frac{n}2\right\rfloor\right) + T\left(\left\...
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15
votes
Accepted
Can any finite problem be in NP-Complete?
If a finite problem is NP-complete then P=NP, since every finite problem has a polynomial time algorithm (even a constant time algorithm).
When we say that Sudoku is NP-complete, we mean that a ...
- 274k
15
votes
There are pretty much no algorithms in P
Actually you're wrong. In the example you give, you take a list of integers, but even with binary representation for the numbers in the list, the input length is still linear in the number of elements ...
- 311
14
votes
Accepted
Give a specific case where calling a polynomial time function n times gives an exponential time algorithm
A concrete example is repeated squaring. Squaring an integer of length $n$ takes time $O(n^2)$ (using the naive algorithm; you can do much better with more complicated algorithms). If you square an $n$...
- 274k
14
votes
Accepted
Proof Complexity of a Proof or Disproof of P = NP
It is known that any proof of super-polynomial circuit lower bounds (which are slightly stronger statements than $P\neq NP$) require super-polynomial, even exponential size proofs in weak proof ...
- 408
14
votes
Why are computability problems always written in full caps?
In the area of discrete mathematics, sets are usually typeset in capital letters. The above problem classes are sets of problems, e.g. SAT is the set of all boolean satisfiability problems.
Thus, the ...
- 1,264
13
votes
Can a subset of an NP-complete problem be in P?
In fact, you don't need the P$\,\neq\,$NP hypothesis, since there are even infinite constant-time decidable subsets of NP-complete problems. For any NP-complete language $L\subseteq\{0,1\}^*$, let $L'...
- 81.4k
12
votes
Accepted
How can I show that the Cook-Levin theorem does not relativize?
Please refer Does Cook Levin Theorem relativize?.
Also refer to Arora, Implagiazo and Vazirani's paper: Relativizing versus Nonrelativizing Techniques: The Role of local checkability.
In the paper ...
- 4,807
12
votes
Accepted
An one-sentence proof of P ⊆ NP
Since L is in P, you can answer the word problem in polynomial time. To show that L is in NP as well, we need to provide a polynomial checking relation $R$ such that
$$ w\in L \Leftrightarrow \exists ...
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12
votes
Accepted
If NP is the class of problems that cannot be solved in polynomial time, what is co-NP?
Your prof was absolutely not rigorous (i.e. completely wrong), that's why the distinction between NP and co-NP doesn't make sense with his definition. Better definition:
Def.: A decision problem (...
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12
votes
Is this possible when it comes to the relations of P, NP, NP-Hard and NP-Complete?
Actually, your version is correct and Wikipedia's is wrong! (Except that it has a tiny disclaimer at the bottom.)
If $\mathrm{P}=\mathrm{NP}$, Wikipedia claims that every problem in $\mathrm{P}$ is $\...
- 81.4k
11
votes
Evolving artificial neural networks for solving NP problems
It seems other answers while informative/ helpful are not actually understanding your question exactly and are reading a little too much into it. You didn't ask if neural networks would outperform ...
- 11k
11
votes
Is Post Correspondence Problem in NP?
Your witness is polynomial in the size of the solution not in the size of the input. You have no way of bounding the length of potential solutions. Your proof shows that PCP is recursively enumerable.
- 851
11
votes
Accepted
Why are Chess, Mario, and Go not NP-complete?
It's a common misconception that chess is NP-hard. Generalized chess may be NP-hard. Chess has an 8x8 board, generalized chess has an nxn board with many pieces.
The question then becomes if ...
- 2,469
11
votes
A problem in NP but not NP-complete?
As written, the question is a bit trivial: if NP = NP-complete, then since P $\subseteq$ NP we get P=NP since every problem in P would be NP-complete.
I suspect what's meant, though, is the following:...
- 2,674
11
votes
Assuming P = NP, how would one solve the graph coloring problem in polynomial time?
If P=NP, that means there is for any given problem in NP, for example, the problem "Is $G$ $k$-colourable?", where $G$ is a finite graph and $k$ an integer, there is an algorithm to solve it in ...
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