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Questions related to the (computational) complexity of solving problems

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0
votes
0answers
12 views

Unrestricted grammar for a^n^2

I have a basic idea of how to generate an unrestricted grammar for a^2^n, a^3^n, or any a^c^n where c = constant. For a^2^n: S -> @aP P -> e P -> RP aR -> Raa @R -> @ @ -> e For a^3^n: S -> ...
1
vote
0answers
9 views

TSP cycle approximation proofs

Having a hard time formulating the proof for this. The professor kinda breezed through the problems and I just want an explanation on how best to tackle this and how one would prove what it's ...
1
vote
2answers
42 views

A basic question about approximation algorithms for the Traveling Salesman Problem

Approximating the traveling salesman problem (TSP) within a constant factor $k$ is hard. The standard proof shows that the existence of such an approximation allows the Hamilton Cycle problem to be ...
3
votes
2answers
45 views

Why is the O(nW) algorithm for the Knapsack problem not a polynomial one?

On the wikipedia page for the knapsack problem it says that the runtime is $\mathcal{O} (nW)$ and goes on to say that this doesn't violate its classification as NP because the input size is related to ...
-3
votes
0answers
64 views

P versus NP Probelm [on hold]

1- Proving that $$P=NP$$ is NP-Hard. 2- Proving that $$P\neq{NP}$$ is in P. Does the reformulation is correct?
0
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0answers
31 views

Tip Jar splitting problem [duplicate]

Having a bit of an issue with this question and deciding which of these situations requires dynamic programming and which are NP-complete. In all of these situations the input is a "tip jar" ...
2
votes
0answers
57 views

Is this modification of the subset-sum problem NP-complete?

Suppose we have input $s_1,\dots,s_n \in \mathbb Z$ and $t \in \mathbb Z$. We want to know if there exist variables $x_1,\dots,x_n$ in which each $x_i=1/2^k$, where $k \in \{0,1,2,3,4,\dots,\infty\}$, ...
2
votes
1answer
46 views

Is ecological bin packing NP-hard?

The ACM Contest Problem 102 (HTML or PDF) can be paraphrased as: Given 3 bins each containing possibly different number of bottles of 3 colors, move the bottles so that there is one color per bin, ...
2
votes
1answer
31 views

Acyclic Graph in NL

From the book The Nature of Computation by Moore and Mertens, exercise 8.9: Consider the problem ACYCLIC GRAPH of telling whether a directed graph is acyclic. Show that the problem is in NL, and ...
1
vote
1answer
50 views

When is splitting a collection coins two ways NP-complete?

Suppose we have a set $D$ of denominations of coins and a our input is a "tip jar" containing some finite number of these coins (e.g., five nickels, a dime and three quarters). In the first two ...
3
votes
0answers
36 views

Natural language processing complexity

Which natural language processing problems are NP-Complete or NP-Hard? I've searched the [natural-lang-processing] and [complexity-theory] tags (and related complexity tags), but have not turned up ...
0
votes
1answer
33 views

Need help to derive the Complexity!

Please help me regarding this one! What will be the communication and computation complexity of this code?? ...
3
votes
3answers
49 views

When problem A reduces to problem B, which problem is more complex?

When discussing complexity classes, when we say that problem $A$ reduces to problem $B$, are we saying that problem $A$ is at least as complex as problem $B$, or the other way around?
-2
votes
0answers
23 views

Draw out TM where 1<=n<=|w| the nth character from the left is equal to the nth character from right

We need to write TM for strings over {a,b,c} for some n, 1<=n<=|w| the nth character from the left is equal to the nth character from right. Can someone help me understand the meaning and how ...
3
votes
2answers
233 views

complexity of determining whether a language given by context free grammar is empty

I know that it is decidable problem to check whether given context free grammar represents empty language -- for instance, AFAIR one could convert it to Chomsky normal form, and then check if any word ...
3
votes
1answer
45 views

What if $NP\subseteq BPP$?

I'm new to complexity and came upon the following exercise which I'm unable to solve. Prove that if $NP\subseteq BPP$ then $\Sigma_2^p=\Pi_4 ^p$.
6
votes
3answers
225 views

What problem cannot be solved by a short program?

BACKGROUND: Recently I tried to solve a certain difficult problem that gets as input an array of $n$ numbers. For $n=3$, the only solution I could find was to have a different treatment for each of ...
1
vote
0answers
58 views

Check whether a directed, rooted spanning tree is actually some shortest-paths tree in $O(V + E)$ time

Given a directed graph $G = (V, E)$, with all edge weights being non-negative, someone has written a program that he/she claims implements Dijkstra's algorithm. For a fixed starting vertex $s$, the ...
0
votes
0answers
45 views

Is there a poly time algorithm for finding all approximate zeros contained in $[0,1]$ of a continuous poly computable $f(x)$ with at most m zeros?

Specifically we know that for our function $f$ ($f$ takes real values), $\infty>f(0)>0$, while $-\infty<f(1)<0$. So the classical binary search will find an approximate zero within ...
2
votes
0answers
37 views

Does $C$-complete = co-$C$-complete imply that $C$ = co-$C$? [closed]

Lets have an arbitrary complexity class $C$. Does $C$-complete = co-$C$-complete imply that $C$ = co-$C$? I think that the answer is yes, but I am not sure whether my reasoning is correct. I tried ...
3
votes
2answers
109 views

Word tiling, where you must use each tile exactly once

Given words $w_1,\ldots,w_n$ in binary alphabet and another word $w$, decide if $w$ can be written as a product $w = w_{i_1} \cdots w_{i_n}$ (in the monoid $\{0,1\}^\ast$) for some permutation of ...
6
votes
1answer
270 views

Proving NP-hardness of strange graph partition problem

I am trying to show the following problem is NP-hard. Inputs: Integer $e$, and connected, undirected graph $G=(V,E)$, a vertex-weighted graph Output: Partition of $G$, $G_p=(V,E_p)$ obtained ...
1
vote
0answers
80 views

Are rational functions with positive integer coefficients honest?

For every rational function $p(x)/q(x)$ where $p$ and $q$ are polynomials with non-negative integer coefficients, does there exist a polynomial function $h$, such that, if you input a reduced fraction ...
1
vote
1answer
71 views

Proving NP hardness of maximum sum of means of a partition into k sets

I am trying to show the following problem is NP-hard and would like some help. Inputs: Integer $k$, and unordered set of $N$ numbers, $O$ Output: the $\max \sum\limits_{S_i \in S} ...
1
vote
1answer
60 views

Prerequisites of computational complexity theory

what's the prerequisite topics needed for understanding computational complexity theory and analysis of algorithm ...including big-O and Big-theta notations and these staff. I want a mathematical ...
3
votes
0answers
210 views

Showing that the language of graphs and nodes on an odd cycle is in NL

Let L be the language containing all the pairs (G,v) where G is a directed graph and v is a vertex in G such that G contains a cycle that contains v and the number of different vertices that appear ...
0
votes
2answers
46 views

NP-hard proof with reduction from two known NP-hard problems

As I understand, to show that a certain problem P is NP-hard we can reduce a known NP-hard problem, Q, to problem in P in polynomial time. To show that the problem P is NP-hard in strong sense, we can ...
4
votes
1answer
55 views

Extracting maximum information from a set of exam answers and their scores

Imagine we have a multiple-choice exam with N questions. Suppose we have a set of K answer sheets to the exam and their total scores (1 for a correct answer on a question, 0 for incorrect). How much ...
8
votes
1answer
151 views

Are all known algorithms for solving NP-complete problems constructive?

Are there any known algorithms that correctly output "yes" to an NP-complete problem without implicitly generating a certificate? I understand that it is straightforward to turn a satisfiability ...
4
votes
1answer
46 views

Computational complexity for more general problems

When I read computational complexity I encounter problems like 3-SAT, set cover, knapsack. In the first two variables are discrete. In knapsack the weights and values are integer and all three ...
0
votes
0answers
16 views

Physically implement Post BQP

Why is it not possible to physically implement post selection in quantum computing? If there were a means to implement it we could solve all PP problems in polynomial time since PP = PostBQP
4
votes
1answer
73 views

Decide whether there exists a walk of weight exactly k

Consider the following problem: Input: a directed graph $G = (V,E,\omega)$ where $\omega : E \longrightarrow \mathbb{Z}$, two vertices $v_1, v_2 \in V$, and a weight $k \in \mathbb{Z}$ Question: ...
3
votes
1answer
55 views

The buckets of water problem

Let's consider the following problem (buckets/pails of water problem) (This problem may be known with different name. If does, please correct me). Let $B=\{b_1,...,b_n\}$ be a set of $n$ buckets. ...
3
votes
4answers
444 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 ...
2
votes
1answer
33 views

Proof of P-Hardness by reduction

I want to proof the P-Hardness of a language. Why is it enough to make a reduction-proof from an other, already P-Complete known language?
3
votes
0answers
58 views

Problems that provably require quadratic time

I'm looking for examples of problem which has a lower bound of $\Omega(|x|^2$) for input $x$. The problem needs to have the following properties: $\Omega(n^2)$ runtime proof for any algorithm - ...
8
votes
1answer
88 views

What is the name of the problem? (partitioning graph into three covers)

I was wondering if this problem has a name: Given a simple graph whose edges are colored red, blue and green, $G=(V,B\cup R\cup G)$, is there a vertex-coloring $c:V\to \{B,R,G\}$ such that every edge ...
0
votes
0answers
13 views

When proving a problem is NP-C, how do I select another NP-C problem for the transformation? [duplicate]

I'm taking an algorithms course in which we are discussing proofs that problems are NP-Complete. Our proofs usually take the form: Given a problem $\Pi$, 1. Prove that $\Pi$ is NP. 2. Select an ...
6
votes
0answers
48 views

Exponential analogue of NC?

Nick's Class (NC) is the class of problems that can be decided in poly-log time using a polynomial number of processors. I want to know about the exponential analogue, which would cover problems that ...
1
vote
1answer
54 views

Is an algorithm in pseudocode a reasonable way to establish complexity?

We define the language $$ L = \{a^nb^n : n\geq0 \} $$ and we want to prove the following $$ L = \mathrm{DSPACE}(\log n)\,. $$ So we have to prove that by using $\log n$ space on the work tape of ...
8
votes
1answer
109 views

Are regex crosswords NP-hard?

I was fooling around the other day on this website: http://regexcrossword.com/ and it got me wondering what the best way to solve it was. Can you solve the following problem in polynomial time or is ...
1
vote
1answer
147 views

Is finding if a graph has k isolated nodes a NP-Complete problem?

I was wondering if finding if a graph has k or more isolated nodes is a NP-Complete problem. I found the following problem: Prove that the following problem is NP-Complete. Given a set of T ...
1
vote
1answer
29 views

Polynomial time optimisation algorithm for a poly-time computable function with bounded number of maxima?

Suppose we have a polynomial time algorithm for computing a function (we think of as existing on rational numbers between $0$ and $1$ of limited binary length n). We know that this function is made up ...
2
votes
2answers
103 views

Finding an exactly weighted st-path in a digraph

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
3
votes
1answer
75 views

Lower space bound on a turing machine accepting palindromes

Let $$ PAL = \lbrace x \in \lbrace 0, 1, \# \rbrace^* | x = rev(x) \rbrace $$ How do I show that a turing machine deciding $PAL$ must use space $\Omega(\log n)$? I have a feeling that I need to use ...
0
votes
0answers
45 views

Do poly-computable differentiable functions on [0,1] with bounded number of turning points have poly-time computable inverse?

Given a polynomially computable continuous function which is a composite of m strictly monotone functions, can we guarantee the existence of polynomially computable inverse? The function I have in ...
-1
votes
1answer
39 views

Relationship between an NP-hard problems with the subsets of them (part 2)? [duplicate]

I asked two questions about NP-hard problems here Relationship between an NP-hard problems with the subsets of them? and here Does this manner of proof for being NP-hard is true? but unfortunately ...
4
votes
1answer
44 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 ...
1
vote
1answer
48 views

What do we know about covering the edges of a graph by disjoint paths?

Two related things I have heard/know of are, (1) That there exists a polynomial algorithm to find a cover of the vertices by $k$ vertex disjoint cycles. (Can someone give a reference for this?) ...
-1
votes
1answer
40 views

How to reduce bin-packing problems? [duplicate]

This is my first time with reductions and I can't figure out how to do them. I have read the few standard examples that are given in the standard books. For example, given $n$ numbers $\{ 0 < ...