Questions related to the (computational) complexity of solving problems

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1answer
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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 ...
2
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1answer
86 views

Log-Space Reduction $CO-2Col \le_L USTCON$

I want to show that $CO-2Col \le_L USTCON$ (Log-Space reduction) $USTCON$ The $s-t$ connectivity problem for undirected graphs is called $USTCON$. [Input]: An undirected graph $G=(V,E)$, ...
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1answer
63 views

what are the basic/typical/common mistakes in P=NP claims? [duplicate]

the P vs NP problem attracts a lot of attention, not all of it desirable, for a wide variety of reasons. there are many P=NP claims eg on this widely cited list maintained by mathematician Woegeorgi, ...
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1answer
50 views

P, NP and polynomial time reduction?

If $P = NP$ would this imply that polynomial time reduction from an $NP$- to a $P$-problem would be possible? And if $P\neq NP$ does it imply that a polynomial time reduction from an $NP$- to a ...
2
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1answer
87 views

The crux of Savitch's Theorem

In "Introduction to the Theory of Computation" by Sipser, Savitch's theorem is explained as an improvement to a naive storage scheme for simulating non-deterministic Turing machines (NTM). I am going ...
5
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1answer
84 views

How to compare algorithms in class NC time complexity with other classes?

I know these relations : \begin{gather} \mathrm{NC}^1 \subseteq \mathrm{NC}^2 \subseteq \dots \subseteq \mathrm{NC}^i \subseteq \dots \subseteq \mathrm{NC} \\ \mathrm{NC}^i \subseteq \mathrm{AC}^i ...
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0answers
79 views

Proof of PCP theorem

I am reading the proof of PCP theorem in Proof Verication and Hardness of Approximation Problems. The following paragraph appears in section 3 (page 4), "Outline of the Proof of the Main Theorem". ...
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0answers
64 views

Assume that $\mathsf{NP} \subseteq \mathsf{P}/\text{log(n)}$, does it imply that $\mathsf{P} = \mathsf{NP}$? [closed]

I am trying to either prove or refute the claim mentioned in the title. Any ideas ?
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1answer
45 views

Relation between digraph and NP-Complete problem

Can there be any relations regarding the number of nodes available in a digraph so that to qualify it as NP-Complete problem. If we consider this problem for instance: Input: A digraph $G=(V,E)$ and ...
4
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1answer
158 views

Is building this tournament fixture an NP-Hard / NP-Complete problem?

I'm curious to know if this problem is NP-Hard / NP-Complete, which I believe would mean I'm unlikely to find a polynomial-time algorithm to solve it. I have written a program which randomly ...
2
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1answer
36 views

Proof of sum of powerset?

Is there already a worst case time complexity proof for the sum of all elements in a power set? I would assume, naively, you have to just add everything, which would run in about 2^n, where n is the ...
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1answer
142 views

Runtime bounds on algorithms of NP complete problems assuming P≠NP

Assume $P\neq NP$. What can we say about the runtime bounds of all NP-complete problems? i.e. what are the tightest functions $L,U:\mathbb{N}\to\mathbb{N}$ for which we can guarantee that an optimal ...
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2answers
75 views

Does a polynomial-time reduction from A to B imply that B is in NP if A is?

Let f be a polynomial-time reduction of a decision problem A to a decision problem B. We know that, if B $\in$ P then A $\in$ P. Similarly, if B $\in$ NP then A $\in$ NP. However, what about the other ...
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0answers
46 views

How to do well in Computational theory courses? [closed]

I'm having a lot of trouble solving problems in my Comp Theory class. I just have no idea how to formulate arguments for certain things like proving the concatenation of two non reg langs can have reg ...
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3answers
191 views

Relationship between Las Vegas algorithms and deterministic algorithms

I'm wondering why the following argument doesn't work for showing that the existence of a Las Vegas algorithm also implies the existence of a deterministic algorithm: Suppose that there is a Las ...
5
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1answer
144 views

Reduce Vertex cover to SAT

I need to reduce the vertex cover problem to a SAT problem, or rather tell whether a vertex cover of size k exists for a given graph, after solving with a SAT solver. I know how to reduce a 3-SAT ...
3
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1answer
53 views

What is the difference between RAM and TM

In case of algorithm analysis we assume a generic one processor Random Access Machine(RAM). As I know RAM is machine which is no more efficient than the Turing machine.All algorithms can be ...
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0answers
17 views

Variants of the 3-Partition problem

The 3-Partition problem (wiki) is a $\text{NP}$-complete problem which is to decide whether a given multiset of integers can be partitioned into triples that all have the same sum. It is well-known ...
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0answers
10 views

Show polynomial hierarchy levels closed under reduction [duplicate]

Most books assume that this is obvious, but I can't see how each $\Sigma_k=NP^{\Sigma_{k-1}}$ level in the polynomial hierarchy is closed under polynomial-time reductions. Is there something that I'm ...
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2answers
76 views

Every language that is reducible to a language in $\Sigma_i^p$ is also in $\Sigma_i^p$ . How?

The complexity class $\Sigma_{k}^{p}$ is recursively defined as follows: \begin{align} \Sigma_{0}^{p} & := P, \\ \Sigma_{k+1}^{p} & := P^{\Sigma_{k}^{p}}. \end{align} Why is every language ...
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2answers
44 views

How to represent a 0-valid boolean formula?

I read in these two papers http://www.ccs.neu.edu/home/lieber/courses/csg260/f06/materials/papers/max-sat/p216-schaefer.pdf and http://people.csail.mit.edu/madhu/papers/noneed/fullbook.ps that if we ...
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1answer
96 views

Relaxed graph coloring, with penalties for assigning adjacent vertices the same color

Consider a set of $N$ nodes. There is a $N\times N$ non-negative valued matrix $D$ where the $(i,j)$th element $d_{ij}$ gives the "positive metric" between node $i$ and $j$, where $i,j\in [N]$. Thus ...
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1answer
93 views

Proving NP-completeness of a graph coloring problem

Given a graph $G=(V,E)$ and a set of colors $k<V$. Find a assignment of colors to vertices that minimizes the number of adjacent vertices in conflict. (Two adjacent vertices are in conflict if they ...
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1answer
24 views

There is equivalence in an NP-hardness proof or not?

I want to show that some problem $P_1$ is NP-hard. I have a problem $P_2$ that is NP-complete. From an instance of $P_2$ I created in polynomial time an instance of the problem $P_1$. My question is: ...
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2answers
89 views

Hardness of mixed 3-SAT and 2-SAT formula

It is well known that 3-SAT is $\sf NP$-complete , but 2-SAT is in $\sf P$. Let there be a formula with $n-1$ clauses with 2 literals each and only 1 clause with 3 literals. We can solve this ...
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2answers
52 views

Do the polynomials in “polynomial time” have integer, real or complex coefficients?

This is probably a very basic question but do the polynomials in "polynomial time" have integer, real or complex coefficients? Everywhere I looked it just says "polynomial expression". I am guessing ...
2
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1answer
94 views

Bin packing problem or not?

Suppose I have $N$ bins and $M$ items as depicted in the figure below (3 bins and 3 items): Suppose that every bin has unit capacity and the weights of the items depend on the bins used. I want to ...
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0answers
55 views

Reduction from Steiner tree to minimum set cover

I am trying to teach myself complexity. I am trying to come up with a reduction from minimum set cover (given a set of items I, and a set S of subsets of I and an integer k, is there a subset S' of S ...
4
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1answer
108 views

Relation of Space and Time in Complexity?

I'm looking for some clarification on some concepts/facts I came across while studying for a class. I was reading the following wikipedia article. The below specific section and statement intrigued ...
2
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1answer
87 views

Is this NP-completeness proof correct?

I want to prove that a problem $P_1$ is NP-complete. Let say that I want to do a reduction from SAT problem. If the instance of problem $P_1$ depends on $M$ and $N$, can I specify the sturcture of ...
2
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1answer
98 views

Why is determining the size of a maximum independent set or a clique in P?

I read that determining the size of the maximum independent set (and also a clique of maximum size) is in P. The versions that find the actual solution are known to be NP-hard. With respect to ...
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2answers
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Recurrence Problem $T(n) = 3T(n/3) + n$ [duplicate]

My question here is dealing with the residual that I get. We are trying to prove $T(n) = 3T(n/3) + n$ is $O(n*\log n)$. So where I get is $T(n) \le cn[\log n - \log 3] + n$. So my residual is $-cn\log ...
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0answers
142 views

Difference between deterministic and nondeterministic universal turing machine

It is known that a nondeterministic universal turing machine (UTM) can simulate another nondeterministic TM with running time $t(n)$ in time $c t(n)$, where $c$ is a constant. It is also known that a ...
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2answers
62 views

Proving 2P2N SAT is NP-Complete

I hope I named this CNF Boolean sentence the correct way. The way I see it, a 2P2N is where each literal appears twice (or at most twice, but we can say twice without loss of generality). I am ...
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1answer
103 views

Does NP-Complete imply non-satisfiability?

I've seen a lot of text concerning the first NP-Complete problem, Boolean Satisfiability. I guess I'm confused concerning the language. It sounds to me as though the problem could be difficult to ...
2
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1answer
57 views

Reducing 3SAT to Triangle Cover Graph

The Triangle Cover Graph problem is this: Given a graph $G = (V,E)$ and an integer $k$, does there exist a set of at most $k$ vertices of $G$ such that every triangle contained in $G$ also ...
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0answers
31 views

Deterministic subexponential algorithm for parity game [closed]

I'm styding this article: http://www.dcs.warwick.ac.uk/~mju/Papers/JPZ08-SIAMJComp.pdf and there is a step not clear for me. In particular : Can anyone help me to understand what is the ...
4
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2answers
181 views

Time complexity of base conversion

EDIT As requested, a single question Why can't arbitrary base conversion be done as fast as converting from base $b$ to base $b^k$ ? There is a big time complexity difference, so I am also ...
3
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1answer
129 views

Asymptotic lower bound on the number of comparisons needed to find the intersection of unsorted arrays

A homework problem in my current CS class asks us to produce a comparison-based procedure for taking (essentially—there are some poorly-specified rules about duplicates) the set intersection of $k$ ...
3
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0answers
68 views

Is finding all valid nets of a polyhedron NP-hard?

Suppose I wanted to find all valid nets of a polyhedron. Is this kind of problem NP-Hard? My guess is that it is. If you were to increase the "complexity" of the polyhedron (maybe this is the number ...
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3answers
66 views

Big O relation between $2^n$ and $2^{2n}$

I know that: If $f(n) = O(g(n))$ , then there are constants $M$ and $x_0$ , such that $f(n) <= M*g(n), \forall n > n_0$ The other, plain English way of defining it is, If $f(n)=O(g(n))$ ...
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2answers
105 views

2 cases for P = NP

As we all know the million dollar question in Computer Science P=NP or not. I was trying to understand it and got some doubts please tell me whether I'm right or wrong N=NP in two cases Case 1: ...
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1answer
163 views

Why is this function computable in $O(n^{1.5})$ time?

My textbook says: "We define the function $f\colon \mathbb{N}\to\mathbb{N}$ as follows: $f(1)=2$ and $f(i+1)=2^{f(i)^{1.2}}$. Note that given $n$, we can easily find in $O(n^{1.5})$ time the number ...
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3answers
231 views

Why can't we flip the answer of a NDTM efficiently?

I read several times that it is not possible to flip the answer of a NDTM efficiently. However, I don’t understand why. For instance, given a NDTM $M$ that runs in $O(n)$, this text (section 3.3) ...
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1answer
33 views

NP hard relation with NP complete

If any problem P is NP complete then if there is a polynomial time reduction of P to another problem R then what can we say about R.Is it NP-hard or NP complete ? From Theory of computation of ...
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1answer
34 views

lexicographic depth-first search complexity class

It seems to me to be incorrect to say that lexicographic DFS is P-complete, since it isn't a decision problem. There is a corresponding decision problem, first DFS ordering, which is known to be ...
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1answer
70 views

Why can't you write the 2-paths problem as a max-flow problem?

This is a follow-up question to this. Consider the 2-paths problem: Given a directed graph $D=(V,A)$ and pairs of vertices $(s_1,t_1)$ and $(s_2,t_2)$, are there paths $P_1 = (s_1,\dots, t_1)$ and ...
1
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1answer
64 views

Is this path finding problem in a 01-matrix NP-complete?

The problem: Input: An $n \times n$ matrix of 0's and 1's, and a position pos of this matrix (i.e. a pair of integers $i,j$ with $1 \leq i,j \leq n$) Output: YES if there exists a ...
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What is an upper bound on formula size when converting 3-SAT to UNIQUE 3-SAT?

What is an upper bound on formula size when converting 3-SAT to UNIQUE 3-SAT? We can use the Valiant Vazirani Therom, also found here (in more detail). Essentially, it is a randomized algorithm that ...
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1answer
81 views

What makes it so difficult to prove P =/≠ NP? — The subset sum issue [closed]

I can't understand or imagine some fact about NP-hard problems. If I understand it correctly there is only one polynomial-time algorithm needed – for whichever NP-complete problem – to ...