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0
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1answer
28 views

Complexity of general polynomial map evaluation is polynomial?

A polynomial map is equal to another polynomial map iff they take on the same values at each point. So this is different from formal polynomials. So since in $\Bbb{Z}_p$, $x^{p-1} = 1$ for all $x ...
0
votes
0answers
19 views

Is the vertex cover problem NP-Hard in general graphs and in P for bipartite graphs? [closed]

Wikipedia says that finding the minimum vertex cover is NP-Hard. However, for bipartite graphs, I can solve the maximum matching problem with Hopcroft-Karp in polytime and then, through Koenigs ...
3
votes
2answers
268 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 ...
2
votes
1answer
53 views

How to optimally seperate a student body?

Students will identify certain students they want to work with. I have therefore decided to split them into two groups where I want to minimize the number of people in Group 1 who want to work with ...
2
votes
1answer
24 views

Polynomial hierarchy intersection

While familiarizing myself with polynomial hierarchy, I have come across a problem of showing $NP^{\Sigma_{k}^{p} \cap \Pi_{k}^{p}} \subseteq \Sigma_{k}^{p}$. By looking at the proof for $NP^{SAT} ...
2
votes
1answer
33 views

Polynomial Hierarchy — polynomial time TM

Consider, for example, the definition for $\Sigma_2^p$ complexity class. $$ x \in L \Leftrightarrow \exists u_1 \forall u_2 \;M(x, u_1, u_2) = 1, $$ where $u_1, u_2 \in \{0,1\}^{p(|x|)}$, for some ...
1
vote
1answer
28 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 ...
1
vote
1answer
47 views

Relation between logspace-uniform circuits and P-uniform circuits

In the book "Computational complexity" of Barak and Arora, on page 112, they state that: Theorem 6.15: A language has logspace-uniform circuits of polynomial size iff it is in P. The proof of ...
0
votes
1answer
39 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 ...
0
votes
1answer
97 views

Does two languages being in P imply reduction to each other?

Given two languages $L_1$ and $L_2$ that are in $\mathsf{P}$, can it be proven that there is a polynomial time reduction from $L_1$ to $L_2$ and vice versa? If so, how? I noticed that if $L_1$ is the ...
1
vote
2answers
139 views

P vs NP: Assuming P = NP

Lets assume $P = NP$. Can we say if every language $L \in P$, then $L \in NPC$? I read $P \subseteq NP$, which means that $L\in NP$. So I know for example, that a language can be $NP \text{ hard}$, ...
1
vote
1answer
107 views

how to solve NFA acceptance problem in polynomial time

I need to show that the language Anfa = {(A,w)| A is an nondeterministic finite automata that accepts w} can be decided in polynomial time. My problem is every solution that I think of requires ...
7
votes
4answers
140 views

Problems that are NP but polynomial on graphs of bounded treewidth

I heard here that the Hamiltonian cycle problem is polynomial on graphs of bounded treewidth. I am interested in examples/references to different problems which is essentially hard but having ...
1
vote
1answer
99 views

Is the set partitioning problem NP complete?

I know that the set partitioning problem defined like this: Given $S = \left\{ x_1, \ldots x_n \right\}$, find $S_1$ and $S_2$ such that $S_1 \cap S_2 = \emptyset$, $S_1 \cup S_2 = S$ and $\sum_{x_i ...
2
votes
1answer
94 views

Time complexity of languages that are polynomial time reducible to NP complete languages

I am wondering if given the time complexity of an NP-Complete problem or at least some information about it for example if $ SAT\in Time(2^{sqrt(n)})$ (hypothetically) could I assume that all ...
3
votes
1answer
157 views

Is the set-partition problem polynomial time reducible to the subset-sum problem?

There are many solutions on the web showing that the subset-sum problem is polynomial time reducible to the set-partition problem. However, during my search, I came across the following powerpoint ...
-2
votes
1answer
69 views

Proving $PH = NP$ [closed]

Show that if 3SAT is polynomial-time reducible to $complement of 3SAT$ then $PH = NP$. Above problem is Exercise problem from Arora and Barak, i don't know how to solve this problem,if anybody knows ...
-1
votes
1answer
133 views

Proving that the complexity class $P$ is closed under union

The following is my proof for $P$ being closed under union. I wish to know if my proof is correct in addition to what it means for the union of two problems. Proof: Let $p_1, p_2 \in P$ Then by ...
0
votes
0answers
87 views

Proving languages are complete on NL?

I'm trying to prove that every language that is not the empty set or {0,1}* is complete for NL (nondeterministic logarithmic space) under polynomial-time Karp reductions. I'm really not sure how to ...
2
votes
1answer
110 views

Is it possible to solve parity game problem in polynomial time?

Is it practically possible or even near possible to make parity game to be solved in polynomial time? If yes, how? and if No, why?
-2
votes
1answer
78 views

Deterministic Random access machine and polynomial time

How do we prove $M$ that is a deterministic random access machine that decides a problem $A$ for an input $i$, and $u_M(i)$ is the set of addresses of those registers that occur at least once with $s$ ...
4
votes
2answers
199 views

Why do most scientists believe that P≠NP?

I read that most scientist don't believe that P=NP. This 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 ...
3
votes
2answers
191 views

Polynomial-time algorithm with exponential space is eligible?

I'm curious about two things. When we define the class called "probabilistic polynomial-time algorithm" in computer science, does it include polynomial-time algorithm with exponential space? For ...
5
votes
2answers
223 views

What exactly is polynomial time?

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 ...
5
votes
1answer
108 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 $ ...
-1
votes
1answer
79 views

Properties of polynomial time many-one reductions

I'm working on old multiple choice exams and would like to know if the following statements are true or false: a) $L_1 \le_p L_2 \le_p L_3 \Rightarrow L_1 \le_p L_3$ b) If $L \in \mathsf{NP}$ and $U ...
1
vote
0answers
25 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 ...
3
votes
2answers
69 views

What is a Turing Machine in class coNP

On the wikipedia article about the polynomial hierarchy http://en.wikipedia.org/wiki/Polynomial_hierarchy it says "$A^B$ is the set of decision problems solvable by a Turing machine in class A ...
0
votes
0answers
47 views

Nash Equilibrium in Tree of Bounded Degree

I have an exercise which I can't solve. Exercise. Consider a game where the players have $2$ pure strategies each and assume that the graph $G$ is a tree with maximum degree $3$. Give a polynomial ...
0
votes
1answer
75 views

Exponential reduction vs Polynomial Reduction

I'm having trouble understanding reduction. Lets say you have a decision problem A that is NP-Complete. Also, another problem B the can be reduced from A. What can you say about B if: 1) The ...
1
vote
1answer
90 views

Prove that if a problem L can be decided in polynomial time, then L ≤p L' for any other problem L'

So we know that there exists a Turing Machine $M$ and a polynomial $T$ such that: $M$ halts on all inputs within at most $T(|x|)$ steps If $x$ is in $L$ then $M$ accepts $x$ If $x$ is not in $L$ ...
2
votes
1answer
38 views

Showing filling a container with rectangles is hard by reducing from SUBSET-SUM

Given a set of rectangles, $D = \{ (a_1, b_1), (a_2, b_2) \dots , (a_n, b_n) \}$, where in each pair $(a_i, b_i)$, $a_i$ represents the height of the rectangle and $b_i$ the width, and given another ...
-1
votes
1answer
105 views

Reducing from Hamiltonian Cycle problem to the Graph Wheel problem cannot be proved vice versa [closed]

I saw a proof by Saeed Amiri, We will add one extra vertex v to the graph G and we make new graph G′, such that v is connected to the all other vertices of G. G has a Hamiltonian cycle if and only if ...
0
votes
2answers
205 views

P is contained in NP ∩ Co-NP?

How should I show that ${\sf P}$ is contained in ${\sf NP} \cap {\sf CoNP}$? I.e., all polynomial time solvable problems and their complements are verifiable in polynomial time.
2
votes
1answer
37 views

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? ...
6
votes
2answers
3k 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 ...
3
votes
2answers
220 views

If A is poly-time reducible to B, is B poly-time reducible to A?

Basically, is the following statement true? $A \leq_p B$ $\rightarrow$ $B \leq_p A$
0
votes
3answers
324 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 ...
2
votes
2answers
114 views

Why does a polynomial-time language have a polynomial-sized circuit?

I wish to understand why P is a subset of PSCPACE, that is why a polynomial-time langauge does have a polynomial-sized circuit. I read many proofs like this one here on page 2-3, but all the proofs ...
2
votes
2answers
156 views

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 ...
4
votes
1answer
333 views

Propositional formula in DNF can be decided in polynomial time?

For a given propositional formula f in DNF, one can decide in polynomial time, if the formula is satisfiable: Just walk through all subformulas (l_1 and ... and l_k) and check, wheter it has NO ...
-2
votes
1answer
144 views

Exponential input and poly-time algorithm

For a list of integers, of size n, where n is exponential, will merge-sort(n), run in poly-time or psuedo poly-time?
3
votes
1answer
117 views

Polynomially related lengths under two different encodings

I'm reading through "Computers and Intractability: A guide to the Theory of NP-Completeness" by Michael R. Garey and David S. Johnson, p. 20 and I came across this ...
7
votes
1answer
145 views

Find small superset of at least k of n given sets

Say we're given $n$ sets and the size of their union is $m$. We would like to construct a small set which contains at least $k$ of the $n$ given sets. Lets assume that $m$ is less than some ...
0
votes
1answer
92 views

Problem with the definition of P

In "Introduction to Algorithms: 3rd Edition" there is Theorem 34.2, which states $P = \{ L \mid L \text{ is accepted by a polynomial-time algorithm} \}$ "Accepted in polynomial-time" is defined ...
6
votes
2answers
225 views

Is SAT in P if there are exponentially many clauses in the number of variables?

I define a long CNF to contain at least $2^\frac{n}{2}$ clauses, where $n$ is the number of its variables. Let $\text{Long-SAT}=\{\phi: \phi$ is a satisfiable long CNF formula$\}$. I'd like to know ...