# 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.

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### The exact relation between complexity classes and algorithm complexities [duplicate]

Are all algorithms which have polynomial time complexity belong to P class ? And P class do not have any algorithm which does have not polynomial complexity ? Are all algorithms which have non ...
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### Functional problem and verifying solutions

Is there a functional problem for which there is an algorithm that can decide if a solution is a solution or not in polynomial time but we can't find a solution in polynomial time? Let FP be the ...
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### $m/p$-equivalence holds after union with an arbitrary finite language

Problem 1: Let $A,B$ be languages over some alphabet $\Sigma$, if $A \equiv_m B$, then for every finite language $C$, $A \cup C \equiv_m B \cup C$. Problem 2: Problem 1 but using polynomial time ...
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### Methods of turning a decision problem into finding the certificate?

I usually find this in the context of asking about NP-complete problems, but any decision problem works. We start by assuming there's a polynomial time algorithm that gives the yes or no answer. If it'...
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### Characterizing the range of a polytime function

Is it true that an infinite language is in P iff it is the range of a length increasing polytime function? I ask because I know that it is a basic result in computability theory that a set is ...
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### GOTO vs. including line in loop - will it affect efficiency?

Let's say I have an algorithm something like as follows: ...
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### If A is in P and B is non-trivial, then A ≤p B [duplicate]

On wikipedia's article on Polynomial-time reduction it states: Every nontrivial decision problem in P (the class of polynomial-time decision problems, where nontrivial means that not every input ...
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### Satisfiable CNFs where each clause contains logarithmically many different literals

Studying for my finals in Complexity theory. This question comes up in different variants and it requires to use probability. A side note before, to be more clear: A CNF clause consists of $n$ clauses ...
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### How is Hypergraph Isomorphism (HI) reduced to Graph Isomorphism (GI) in polynomial time?

This question states that the problem of Hyper-graph Isomorphism is equivalent to Graph isomorphism. I have not been able to find a description of the reduction so I am wondering how that might work ...
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### How to prove an polynomial run time is faster than exponential using definition of big O

This is for homework so feel free to not give me an answer but steer me in the right direction. The problem states: Prove that $n^{1000000} = O(1.000001^n)$ using the formal definition of Big-O. ...
203 views

### Are finite-domain binary constraint satisfaction problems solvable in polynomial time?

Suppose a CSP has $n$ variables with finite domains of maximal size $d$. Furthermore, all constraints on the variables are binary. Can such a CSP be solved in polynomial time in $n$ and $d$? This was ...
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### Can you convert a positively weighted DAG into a non-weighted DAG in polynomial time?

Given a positively weighted DAG (directed acyclic graph) $D = (V,E)$, can you create a new non-weighted DAG $D'$ by converting each edge with weight $w(e) = x$ into x non-weighted edges and vertices? ...
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### How undecidable is it whether a given Turing machine runs in polynomial time?

The proof of Theorem 1 that PTime is not semi-decidable in this recent preprint effectively shows that it is $\mathsf{R}\cup\mathsf{coR}$-hard. The proof itself is similar to undecidability proofs at ...
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### Polynomial Time Algorithm for Steiner Tree Problem

I know about Steiner Tree Problem. It is stated as Input to Steiner Tree Problem is a weighted graph G and a subset T of the nodes (called terminal nodes) and goal is to find a minimum weight tree ...
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### Are you allowed to change the specifications of a problem when doing reductions?

I'm doing a polynomial time reduction from problem A (known graph problem) to problem B (funky and specific longest path problem). There is a lot of demands on how problem B is supposed to be solved. ...
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### Is there a “well known” example of a constraint satisfaction problem on a 3-element set which is polynomial-time solvable?

I'm basically looking for an example (in maybe graph theory) of a constraint satisfaction problem which has a 3-element set as a domain and the problem is known to be polynomial-time solvable.
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### Complexity of polynomial interpolation

Polynomial Interpolation in the general case is $O(n^2)$ time complexity, but it can be done better in particular situations. For instance, when the polynomial can be evaluated at the complex roots of ...
531 views

### $DTIME(f(n)) \subset of DSPACE(f(n))$

I think this is again an easy one: $DTIME(f(n)) \subset DSPACE(f(n))$ They say its trivial but I dont see it, why? And would $DTIME(f(n^2)) \subset DSPACE(f(n^2))$ also be true? if yes why ...
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### Show that the SAT Problem for CNF formulas with at most two occurences of each variable can be solved in polynomial time

Assuming, I have an arbitrary CNF Formula in which each variable has at most two occurences, how can I proof/show that this can be solved in polynomial time? My first thoughts so far: because each ...
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### What is the precise definition of pseudo-polynomial time (feat. Counting Sort)

From wikipedia In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the length of the input (the number of bits required ...
TQBF consists of alternating quantifiers, so does $\Sigma^2_n$ for fixed $n$. So given a formula in TQBF, shouldn't there be a level of the polynomial hierarchy that solves it? I think this is ...
Using the following definition: Reduction: There is a polynomial-time reduction from problem $X$ to problem $Y$ if arbitrary instances of problem $X$ can be solved using: Polynomial ...