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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326 views

P/NP - Polynomial Reduction vs Certificate

I am learning about the P/NP problem right now, and I don't understand when to use polynomial reduction and when to use a certificate. How I understand polynomial reduction is that you can use it to ...
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24 views

Use Fact to decompose a given number

Let Fact be the following decision problem. Given two natural numbers $k \le n$ the machine accepts if and only if $n$ has a non-trivial divisor that is less than or equal to $k$. Prove that if Fact $\...
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168 views

Simple Hamiltonian cycle reduction

HAMPATH Input: An undirected graph $G$ and 2 nodes $s, t$ Question: Does G contain a Hamiltonian path from $s$ to $t$? HAMCYCLE Input: A undirected graph $G$ and a nodes $s$ ...
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171 views

Polynomial Time reducible explanation

Have a set of examples given to me, but I'm pretty sure they're all wrong. Can someone verify that my understanding of them is correct? If set $Y$ can be solved in $O(2^n)$ and $Y \leq_p X$ then $X ...
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1answer
42 views

Determine whether a system of $n$ linear equations has solutions in $\{0, 1\}^n$ in polynomial time

I'm trying to determine whether it is possible to decide if a system of $n$ linear equations with integer coefficients and $n$ variables has a solution in $\{0, 1\}^n$ in polynomial time. ...
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122 views

Find Hamiltonian cycle in polynomial time

I want to know for what types of graph it is possible to find Hamiltonian cycle in polynomial time. It would be helpful also to show why on some types of graph finding Hamiltonian cycle would be only ...
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1answer
29 views

How to prove that $n^d$ is $O(b^n)$ from $n$ is $O(2^n)$, given that $d>0, b>1$? [duplicate]

I'm reading Rosen's Discrete Mathematics and Its Application, at Page 212, it's about the "Big-O" notation using in computer science. This is the description in the book: And here is my reasoning: ...
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1answer
404 views

Poly-time reduction is not antisymmetric

Lemma. (Transitivity) "$\leq_p$" is a transitive relation on languages, i.e., if $L_1 \leq_p L_2$ and $L_2 \leq_p L_3$, then $L_1 \leq_p L_3$. Proof. By definition, there are poly-time ...
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1answer
529 views

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 ...
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1answer
51 views

If the difference between two oracles is negligible, is the difference between a PPT algorithm with these two oracles also negligible?

We say a negligible function is a function $\epsilon(n):\mathbb{N}\rightarrow \mathbb{R}$ such that for every positive integer $c$ there exists an integer $N_c$ such that for all $n > N_c$, $$\...
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1answer
60 views

For which level of PH is $\operatorname{VALID}$ complete

I have the decision problem $\operatorname{VALID}$ which is the set of all valid propositional formulas (tautologies), I know that $$\overline{\operatorname{SAT}}\equiv_m^p \operatorname{VALID}.$$ ...
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654 views

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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48 views

On different characterizations of $\mathsf P$

In here it was clarified that $\cap_{f(n)\in\omega(1)}\mathcal C(n^{f(n)})\subsetneq\cap_{\epsilon>0}\mathcal C(n^{n^\epsilon})$ where $\mathcal C(t(n))$ is the class of problems solvable in $O(t(n)...
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84 views

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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1answer
65 views

How to reduce constrained proofs to 0-1 IP

Consider the following problem: Can $X$ be proven in fewer than $Y$ steps, from axioms $Z$, with finitely many transition rules $\tau$? This lies in $NP$, since if I supply a proof $M$, and ...
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1answer
67 views

Is P closed under subwords? [closed]

Given a language $L\subseteq \Sigma^*$ in $P$, is the language $subwords(L) = \{v\in\Sigma^* : \text{there exist } u,w\in \Sigma^* \text{ with } uvw\in L\}$ that consists of all subwords of words ...
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1answer
806 views

What's the difference between “polynomial time Turing-reducible” and “polynomial time many-to-one reducible”? [duplicate]

The following definitions are from Li, M., & Vitányi, P. (1997). An introduction to Kolmogorov complexity and its applications (2nd ed.), pg. 38. A language $A$ is called polynomial time Turing-...
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67 views

Can we decide if a number is a power of any given $K$ in polynomial-time?

It is simple to decide powers of 2 in $O(n)$ time because it's just "0-bit Unary" after bit-1. (eg. $1000$ is a power of 2 in binary). I haven't found many other trivial powers of $K$ that ...
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26 views

Seeking guidance on what to read for Feasibility Binary IP with ''almost total unimodular'' (-1, 0, 1)-Coefficient Matrix and No Obj Function

I am working on an algorithm in graph theory which I wish to prove it's polynomiality/NP-hardness. I am investigating a binary variable (0, 1) integer program which has the coefficient matrix ...
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1answer
51 views

building a polynomial algorithm that solves SAT when given a polynomial TM that solves SAT on two formulas

Here's the question: Assume there exists a polynomial time machine $M$ that receives two formulas $\varphi_1,\varphi_2$ and satisfies the following: If $\varphi_1 \in \mathrm{SAT}$ and $\...
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191 views

0-1 knapsack without repetition

My question is why O(nW) at the knapsack problem is pseudo-polynomial. I read lots of the explanation at stackoverflow, But I don't really understand it. (https://stackoverflow.com/questions/19647658/...
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27 views

Looking for some references on voting theory

After reading through this paper on optimizing the sum of sigmoid functions, http://www.web.stanford.edu/~boyd/papers/pdf/max_sum_sigmoids.pdf, I am interested in the problem addressed in section 7.3 ...
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153 views

Min-Ones-2-SAT getting to vertex cover

In the Min-Ones-2-SAT problem, we are given a 2-CNF formula φ and an integer k, and the objective is to decide whether there exists a satisfying assignment for φ with at most k variables set to true....
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38 views

Smallest string containing sets of letters

I am looking for a solution to this problem: Given multiple sets of letters (Set0={a,b,c,d}, Set1={d,e,f,g}, Set2={a,b,e,g}, ...), what is the minimal length of the string containing all the sets. The ...
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1answer
234 views

CRC computation speed vs polynomials features

I tried to find information about how features of a CRC polynomials influence computation speed of implementations. It is obvious that (depending from the CPU architecture the algorithm runs on) ...
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55 views

Finding simple min-weight path between two vertices in graph with negative edge weights

Given a weighted graph (negative weights are allowed) and two vertices $u$ and $v$, can we find the simple min-weight path between $u$ and $v$? There can be a negative cycle on the path from $u$ to $v$...
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22 views

Sparse feasible solution $|x|_0\le k$ for system of linear inequalities $A x \le b$

Suppose the set of linear inequalities $Ax\le b$, in which $A\in\mathbb{R}^{m\times n},x,b\in\mathbb{R}^n$ is given. Is it possible to determine in polynomial time with regard to $m$ and $n$ if there ...
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1answer
31 views

polynomial time reducibility - $L_{2} \notin \textbf{P}$ and $L_{1} \leq_{p} L_{2} \implies L_{1} \notin \textbf{P}$

If we have two languages $L_{1} \subseteq \Sigma^{\ast}_{1}$ and $L_{2} \subseteq \Sigma^{\ast}_{2}$ I proved that when $L_{2} \in \textbf{P}$ and $L_{1} \leq_{p} L_{2}$ then $L_{1} \in \textbf{P}$ ...
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78 views

Must every NP-Complete Problem have a class of instances which is solvable in Poly time? [closed]

Is there any theorem that states that any NP-Complete Problem has a class of instances solvable in Poly time? For example, some problems like vertex cover are NP-Complete on general graphs but can be ...
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3answers
5k views

What are the implications of P=NP? [duplicate]

Is there a list of implications of $P=NP$? Presumably, a proof of $P \ne NP$ will be by contradiction, for which a list of consequences of $P=NP$ would be useful.
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98 views

What are widely-used, practical applications to come from the study #P problems?

When, beyond theoretical exercises, do we care how many solutions we can find for something? I had an analogous question for TMs before - why is it useful to study machines that can only deliver ...
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35 views

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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35 views

Besides practical computing applications, is there a reason polynomial is “good” and exponential is “bad”? [duplicate]

The overarching theme of computer science seems to be that polynomial time or space or what-have-you for an algorithm is a success, and exponential is a failure. The definitions of P and NP revolve ...
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84 views

Suppose P = NC - what then? [duplicate]

Suppose tomorrow someone discovered a proof that P = NC. What would the consequences for computer science research and practical applications be in this case?
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33 views

Mixing time of three particle systems

Is there anything known about mixing time of Markov chains for three particle systems? It is proved here http://www.ams.org/journals/tran/2005-357-08/S0002-9947-05-03610-X/S0002-9947-05-03610-X.pdf ...
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34 views

Is this problem in P: Finding a common key for a collection of systems of equations?

Let $B=\{b_1=g_1,\cdots,b_n=g_n\}$ be a set of binary variables $b_i$ and their corresponding values $g_i \in \{0,1\}$. Let $M=\{\sum_{e \in A}e \;:\; A \subset B\}$, i.e., $M$ is the set of all ...
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601 views

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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628 views

Polynomial-Time Reduction

I have read many resources, but I cannot understand what the polynomial-time reduction is. In everywhere, this is explained with standard-pattern sentences. Please can anyone explain it in detailed ...
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40 views

What types of string properties are verifiable in polynomial time?

When given the string and the property in question as a potential certificate. Is there any classification theorem that says something along the lines of: all properties (of strings) that have this ...
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1answer
1k views

If the Clique-k Problem is in P, why not Clique as well?

I have looked at the other answers to this but I still don't get it. (for instance: Why is the clique problem NP-complete?) The general clique problem is defined as $\text{CLIQUE} = \left\{ (G, k) | ...
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26 views

Is there a polynomial time algorithm for this decision problem?

Is there a factor in $M$ that is $>$ $1$, but $<$ $M$ that is NOT a factor of $N$? False Result Example $N$ = 8 $M$ = 16 1, 2, 4, 8, 16 There is no integer that is NOT a factor of $N$ that ...
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1answer
59 views

Understanding integer factorization is NP [duplicate]

I can see that Integer factorization problem is in NP. I am looking for a simple intuition behind this. For example if we take the problem of sorting the complexity is $n\log n$ for merge/quick sort ...
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2answers
5k views

Time Complexity of k-clique problem with fixed k [closed]

My question expands on a related question on the link, Why is the clique problem NP-complete? In that post the author argued that while the $k$-clique problem is NP-complete; for a fixed $k$ the $k$-...
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1answer
57 views

Given arbitrary integers of $K$ and $M$, can deciding $2^K$ + $M$ is a prime be in $P$?

Given arbitrary integers of $K$ and $M$, is $2^K$ + $M$ a prime? ...
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76 views

“Given an algorithm, decide whether it runs in polynomial time” is this problem in NP?

This problem is not decidable (reducible to halting problem) but is semi-decidable and therefor verifiable (as those two definitions are equivalent: How to prove semi-decidable = verifiable?). However,...
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1answer
56 views

Do all languages in $P$ have polynomial proofs that they are in $P$?

A proof for a language $L$ belonging to a complexity class $C$ can be framed as there existing a verifier $V$ that accepts this proof as the first part of their input and the language as the second. ...
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27 views

Does the language defined in the details in NP-C or P?

It's known that: $$ \textrm{CLIQUE} = \{(G,k): \mbox{G has a clique of size } k\} $$ is $\textrm{NP-C}$, but what if every vertex has 2 neighbours (as defined in $\textrm{2d-CLIQUE}$)? $$ \textrm{2d-...
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81 views

is $x^{100000000000}$ a “polynomial time”?

per this post $t = x^2$ means the problem is solvable in "Polynomial" time. per this post in the form $$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} { > \boldsymbol{=0}}$$ ...
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1k views

Class P is closed under concatenation

Proving that Class P is closed under concatenation. The answer is given below: But I do not know why stage 2 is repeated at most O(n), could anyone explain this for me please?
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270 views

Is there a poly-time algorithm for expanding out polynomials

so I've been looking around and haven't seen this before. Basically I'm working with a problem in which I need to expand/FOIL out. Something in the form of $$ z = (x+y)(x-y) \implies x^2+xy-xy+y^2 $$ ...

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