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# 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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13k views

### Why is linear programming in P but integer programming NP-hard?

Linear programming (LP) is in P and integer programming (IP) is NP-hard. But since computers can only manipulate numbers with finite precision, in practice a computer is using integers for linear ...
43k views

### What exactly is polynomial time? [duplicate]

I'm trying to understand algorithm complexity, and a lot of algorithms are classified as polynomial. I couldn't find an exact definition anywhere. I assume it is the complexity that is not exponential....
3k views

### 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 ...
1k views

### 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 ...
23k views

### Finding shortest and longest paths between two vertices in a DAG

Given an unweighted DAG (directed acyclic graph) $D = (V,A)$ and two vertices $s$ and $t$, is it possible to find the shortest and longest path from $s$ to $t$ in polynomial time? Path lengths are ...
615 views

### If $n^{\log n}$ is not polynomial or exponential, then what this function is called?

I just found this sentence on page 6 of Garey and Johnson's "Computers and Intractability". Any algorithm whose time complexity function cannot be so bounded is called an exponential time algorithm ...
2k views

### Problems conjectured but not proven to be easy

We have many problems, like factorization, that are strongly conjectured, but not proven, to be outside P. Are there any questions with the opposite property, namely, that they are strongly ...
239 views

### Problems that feel exponential but are P

I'm trying to build a list of algorithms/problems that are "exceptionally useful", as in, solving problems that 'seem' very exponential in nature, but have some particularly clever algorithm that ...
1k views

### Does linear programming admit a strongly polynomial-time algorithm?

The linear programming problem: find a strongly-polynomial time algorithm which for given matrix A ∈ Rm×n and b ∈ Rm decides whether there exists x ∈ Rn with Ax ≥ b. I know that Steve Smale's lists ...
810 views

### Why do we say that polynomial time is easy? [duplicate]

For years, I've been told (and I've been advocating) that problems which could be solved in polynomial time are "easy". But now I realize that I don't know the exact reason why this is so. ...
276 views

### P vs NP and the Time Hierarchy

Assuming $P\neq NP$, is it possible that there exists a $k$ such that $P\subseteq\textsf{NTIME}(t^k)$? There reason I ask this is that I assume the following: P=NP \implies \forall k\ \exists j.\ \...
1k views

### Any problem solved by a finite automaton is in P

After my Theory of Computation class today this question popped in my mind: If a problem can be solved by a finite automaton, this problem belongs to P. I think its true, since automata recognize ...
697 views

### Why do most scientists believe that P≠NP?

I read that most scientists don't believe that P=NP. It might be subjective but can you simplify why not? I'm not informed enough to have an opinion but I'd like to know the definitions and some "...
416 views

### 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 ...
8k views

### 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?
431 views

265 views

### 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 ...
439 views

### Complexity of Linear Diophantine equations

My question is simply, can linear Diophantine equations be solved in polynomial time? Specifically, I am looking at equations of the form $a_1 x_1+a_2 x_2 + ... + a_n x_n = k$, where $a_i,x_i,k$ are ...
659 views

### What is the utility of proving P=NP if we can't find an algorithm that can solve any NP problem in polynomial time?

Here we see a very interesting attempt to show that $\mathrm{P} \ne \mathrm{NP}$ by Norbert Blum. Here we see 116 previous attempts at solving P vs. NP. Here we see the P vs NP problem defined as: ...
122 views

### Time complexity of art gallery-like problem

Suppose that $G = (V,E)$ is a directed graph such that each vertex in $V$ is in at least one edge in $E$. We'd like to decide whether or not $w$ watchmen can be placed on $w$ distinct vertices in $G$ ...
702 views

### 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 ...
218 views

### Finding a perfect matching using an LP

I have a basic question about the power of Linear Programming that has been bothering me for some time. I believe there is something simple I am missing. Linear Programming is $\mathsf{P}$-complete, ...
3k views

### Are all languages in P?

Are all languages in $\mathbf{P}$? Note: The definitions of all the symbols and functions here follow the document . The following is my attempt to answer the question. Assume that we design a ...
33k views

### How to check whether a graph is connected in polynomial time?

I have to solve the following problem: Consider the problem Connected: Input: An unweighted, undirected graph $G$. Output: True if and only if $G$ is connected. Show that Connected ...
476 views

### 3SAT analogous problem in P

Is there a problem like 3 SAT like problem in P where if we find an algorithm for this problem, we can solve all problems in P? For instance if we solve this problem in P, may be we can solve prime ...
2k views

### How can an algorithm have exponential space complexity but polynomial time complexity?

For enumerating the minimal feedback vertex sets of a graph Schwikowski and Speckenmeyer show an algorithm "GENERATE-MFVS" in their publication "On enumerating all minimal solutions of feedback ...
571 views

### 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 ...
1k views

### What is the decidable language in $P/poly$ but not in $P$?

Except for the undecidable unaries I have no idea if there is anything in the gap between $P/poly$ and $P$
59 views

### Given $k$ points in $n$-dimensions, such that $n\geq3$, is there a polytime algorithm for finding a curve that splits them into 2 sets of points?

So in this math exchange question I asked, it was proven that for $n>2$ dimensions, you can always find a curve that separates $k$ points in $n$-dimensional space into $2$ arbitrary sets that you ...
519 views

### Polynomial time algorithm for finding two or more vertex-disjoint cycles

The cycle detection problem for a directed graph has well-known polynomial time solutions, graph traversal algorithms such as Dijkstra algorithm can be used to find whether or not a cycle exists in a ...
182 views

### Proving that a language is not in P using diagonalization

Pardon me if i'm missing something which is very obvious here but i cant seem to figure it out. $E=\{ \langle M, w \rangle \mid \text{ Turing Machine encoded by$M$accepts input$w$after at most$ ...
166 views

### Implicit complexity and interpretation of total languages

In implicit complexity theory we construct languages that characterize what can be computed in various complexity classes. One major result is Bellantoni and Cook where they show that $FP$ can be ...
353 views

### Understanding the Sipser-Gacs-Lautemann theorem

The class $BPP$ contains all the languages decided by a probabilistic Turing machine in polynomial time with probability of success more that 2/3 for every input. The class $\Sigma^p_2$ contains all ...
282 views

### Algorithm for a special case of SAT/#SAT

Does anyone know of an algorithm that can solve the following special case of SAT in polynomial time? Are there any algorithms that can solve the counting (#SAT) version of it in polynomial time? ...
77 views

4k views

### How to prove that problem is not in P

Given some abstract problem how can I prove that this problem is not in P. I mean, what is the method for proving such thesis?
521 views

### How to determine if a black-box is polynomial or exponential

I have a problem which essentially reduces to this: You have a black-box function that accepts inputs of length $n$. You can measure the amount of time the function takes to return the answer, but ...
290 views

### What are the examples of problems which first had large polynomial time complexity algorithms but later the complexity was reduced significantly?

Arora-Barak says It has also happened a few times that the first polynomial-time algorithm for a problem had high complexity, say $n^{20}$, but soon somebody simplified it to say an $n^5$ time ...
590 views

### What is difference between nondeterministic polynomial time and exponential time?

I am not very into computer science theory but i feel like people are defining nondeterministic polynomial time as if it is another name of exponential time. I would be happy if you clarify it. thank ...
### Where/how did a $\log(n)$ factor disappear from well-known algorithms?
Consider the binary search problem on a sorted array containing $n$ integers on 16 bits. Everybody agrees that the binary search needs $O(\log(n))$ time, because it makes at worst $O(\log(n))$ steps. ...