# Questions tagged [polynomial-time]

Use for algorithms, algorithm-analysis and complexity-theory questions that aim for polynomial running time resp. time complexity. Such questions often are are reference-requests or about runtime-analysis or time-complexity.

252 questions
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### relation between ntime and dtime

Given DTIME($n^2$) contains NTIME($n^{100}$) show that P=NP. I think it's supposed to be straightforward but I just can't see it. Take $L$, a language in NP. $L$ has a Turing machine which runs in ...
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### sequence of problems that take $\Theta(n^k)$ for increasing $k$?

Do we know an infinite sequence of decision problems where the most efficient algorithm for each problem takes $\Theta(n^k)$ time, where $k$ increases unboundedly? Suppose for example that we would ...
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### “P may collapse” vs. Time hierarchy theorem

https://en.wikipedia.org/wiki/P_versus_NP_problem states: If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. They further state that this may be ...
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### 2-clause satisfiability associated graph

A 2-clause is a clause with at most two propositions (clauses?) : $(p \wedge q,\neg p \wedge q, \neg p,...)$. I have to show that the folllowing problem is $\in$ P: 2-SAT: Input : A conjunction $\Phi$...
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### Is the MinMax/optimization/search variant of a decision problem always easier/equal?

Is the MinMax/optimization/search variant of a decision problem always easier/equal in complexity because we can always reduce them to their decision variant? From Wikipedia: If the longest path ...
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### Shorten Length Reduction

I've stumbled upon this Question: We say that a reduction $f$ of a language $A$ to a language $B$ is a Shorten length reduction, if there exists a number $n\in N$ s.t for every $w\in A$, s.t ...
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### What is an example for a decidable language not in P?

I'm having trouble showing that $P\neq R$. Obviously $P\subseteq R$, but is there a decidable language which is definitely not (under all answers to open questions s.t. $P=NP$ or $NP=PSPACE$) in $P$ ?...
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### Equality of cardinality of maximum matching and minimum vertex cover in general

I'm preparing for exam and I came across this problem: We say that a graph is König's graph when the sizes of its minimum vertex cover and maximum matching are equal. Find a polynomial time ...
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### Resource bounded reductions for RE-Complete problems

Given that the halting problem is RE-Complete, we can reduce any problem in RE to an instance of the halting problem. Are there are any results on the time-bounds for this reduction? Can we do this ...
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### Are there any RE-complete languages w.r.t. polynomial reduction?

I need to decide if there exists $L\in RE$ so that for every $L'\in RE$ we have $L' \leqslant_p L$, meaning a polynomial-time reduction. I've tried to use $L=A_{TM}$ (the accepting problem), but got ...
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### Show that P is closed against the Kleene star

I have that question that looks kinda easy at first but it is quit hard. Let $L\in P$. Prove that $L^*\in P$ my approach: I tried to generate a turing machine but I got stuck with the thing of ...
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### Does P=NP imply polynomial solutions to #P?

Is it true that $\#P$-complete problems could possibly be solved in polynomial time if P=NP? I know that even some counting problems related to polynomial time decision problems are $\#P$-complete, so ...
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### How do we know for sure that EXPTIME ≠ P?

I'm a beginner in learning about computational complexity and this has stumped me. I've read that by the time hierarchy theorem, it's known that EXP-complete problems are not in P. (Wikipedia) It ...
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### Why is the set of perfect squares in P?

I am reading an article by Cook [1]. In it he writes: The set of perfect squares is in P, since Newton's method can be used to efficiently approximate square roots. I can see how to use Newton's ...
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### Are all languages in P?

Are all languages in $\mathbf{P}$? Note: The definitions of all the symbols and functions here follow the document [1]. The following is my attempt to answer the question. Assume that we design a ...
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### How fast can one compute the power of a number?

Let $x \in \mathbb{R}$ and $k \in \mathbb{Z}^+ \cup \{0\}$ then how fast can one compute $x^k$? If $x, k \in \mathbb{Z}$ then I guess this previous discussion already settled that, How many ...
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### Isn't polynomial identity testing over arithmetic *expressions* trivial?

Polynomial identity testing is the standard example of a problem known to be in co-RP but not known to be in P. Over arithmetic circuits, it does indeed seem hard, since the degree of the polynomial ...
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### Why not to take the unary representation of numbers in numeric algorithms?

A pseudo-polynomial time algorithm is an algorithm that has polynomial running time on input value (magnitude) but exponential running time on input size(number of bits). For example testing whether ...
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### P is closed under power of integer

I'm new in this area of complexity and I'm trying to get into it by understanding basic proofs. I want to prove that if $L\in P$, so $L^k\in P$, where $k$ is non-negative integer. How to prove it in ...
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### Pedagogic reference on cut generating functions

Can you recommend an introduction to the topic of cut generating functions? I am looking for introductory or review-like material. I did find the following survey paper, but it seems to be addressed ...
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### Can the isomorphic graph problem be solved in deterministic polynomial time?

Here is a recent homework problem of mine: Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = {⟨G,H⟩| G and H are isomorphic graphs}. Show ...
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### Can we show that non-determinism adds no power, for some specific running time?

$NP = \cup_{k \in \mathbb{N}} NTIME(n^k)$ $P = \cup_{k \in \mathbb{N}} TIME(n^k)$ Can we show that $NTIME(n^k) = TIME(n^k)$ for a specific $k$? For how large of a $k$ can we show the above ...