Questions tagged [decision-problem]

A question in some formal system with a yes-or-no answer.

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Is there an efficient algorithm for expression equivalence?

e.g. $xy+x+y=x+y(x+1)$ ? The expressions are from ordinary high-school algebra, but restricted to arithmetic addition and multiplication (e.g. $2+2=4; 2.3=6$), with no inverses, subtraction or ...
3answers
5k views

Do any decision problems exist outside NP and NP-Hard?

This question asks about NP-hard problems that are not NP-complete. I'm wondering if there exist any decision problems that are neither NP nor NP-hard. In order to be in NP, problems have to have a ...
2answers
2k views

2answers
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3-SAT where variables occur equally many times as a positive literal and as a negative literal

Let $\phi$ be a 3-CNF formula over variables $x_1,x_2,\ldots,x_n$. Every variable $x_i$, $i \in [n]$, occurs equally many times as a positive literal and as a negative literal in $\phi$. Is it NP-...
3answers
999 views

Does every computational problem have a decision version?

Is the following claim correct?: Every computational problem has a decision version of roughly equal computational difficulty. If the above claim is correct, please give a reference for it. (I ...
2answers
365 views

primitive recursive functional equivalence

Given two primitive recursive functions is it decidable whether or not they are the same function? For example lets take sorting algorithms A, and B which are primitive recursive. While there are many ...
2answers
3k views

Optimization problem vs decision problem - reduction

Assume we have an optimization problem with function $f$ to maximize. Then, the corresponding decision problem 'Does there exist a solution with $f\ge k$ for a given $k$?' can easily be reduced to ...
1answer
124 views

Questions on Graph and Hamiltonian [closed]

From this book and other study in complexity theory, I have seen the following statement: The definition of NP is not symmetric with respect to yes-instances and no-instances. For example, it is ...
1answer
600 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 ...
1answer
1k views

Can the decision version of an optimization problem in NP, be in P?

It is well known that a optimization problem can be turned into a decision problem with an extra parameter: e.g. in TSP we are looking for the lowest cost for a tour, the decision version therefore ...
1answer
524 views

How exactly does a Max 2 Sat reduce to a 3 Sat?

I've been reading this article which tries and explains how the max 2 sat problem is essentially a 3-sat problem and is NP-hard. However, if you see the article, I'm not able to understand why, after <...
2answers
2k views

Decision version of the traveling salesman problem and NP-hardness

Wikipedia says: The problem has been shown to be NP-hard and the decision problem version ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-...
1answer
673 views

Is equivalence of a CFG and an RG undecidable?

I know that the equivalence of two context-free grammars is undecidable, but what about the equivalence of a regular grammar and a context-free grammar?
2answers
153 views

A special case of subset sum

I came across the following problem in my complexity-theory course: Given a set of numbers $A := \{a_1, \dots, a_n\} \subset_{\mathrm{finite}} \mathbb{N}$ and a number $b$ also in $\mathbb{N}$ such ...
2answers
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Is it decidable that a context free language contains a given regular language?

I've been asked to solve this problem, but I'm completely stuck now. Is the set $\{G \in\text{CFG} \mid L(G)\supseteq L(A) \}$ where A is DFA fixed beforehand decidable? I know I've to find a ...
1answer
2k views

Decision Problem vs. Optimization Problem [duplicate]

Is the following statement correct? If a decision problem is NP-complete, the corresponding optimization problem can not be solved in polynomial time.
1answer
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Obtaining an acyclic graph by removing edges using an algorithm that decides ACYCLIC

i don't understand the following: If there's an algorithm that can decide ACYCLIC in Polynomial time, then there's an algorithm who returns a set of k edges, so that the graph obtained by deleting ...
7answers
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Is legislation NP-complete?

I would like to know if there has been any work relating legal code to complexity. In particular, suppose we have the decision problem "Given this law book and this particular set of circumstances, is ...
4answers
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2answers
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Is deciding if there's a solution to a single multivariate quadratic equation NP-hard?

I know that given a system of multivariate quadratic equations (i.e, of the form $x^T Ax+b^T x=c$), deciding if there's a solution is NP-hard. Is deciding if there's a solution to a single ...
1answer
2k views

Reduction: Vertex Cover to Binary Integer Program (Decision Problem)

I am stuck with the following task: Show that the Decision Problem "Vertex Cover" is polynomial-time reducible to the Decision Problem "Binary Integer programming". I have the feeling that there must ...
2answers
575 views

Why are decision problems easier than the equivallent optimization problems?

Suppose that we have an optimization problem defined as follows: $OPT$ = Given an input string defining a set of feasible solutions $F$ and an objective function $f$, find $x\in F$ maximizing $f(x)$ ...
2answers
634 views

Do all algorithms aim to solve decision problems?

An algorithm is a āunambiguous specification of how to solve a class of problems... calculation, data processing and automated reasoning tasksā Are the class of problems an algorithm can solve more ...
0answers
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Is there a generic procedure to produce (hard enough) decision problems?

This comment by @DerekElkins suggests a general method of constructing decision problems for problems with bit-strings as output, of which a slightly formalised version is the following: Given a ...
2answers
642 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 ...
1answer
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Is there any interesting consequence of $\mathrm{DLogTime}$-uniform ${\mathrm{Mod}_6}^0=\mathrm{NP}$

$\mathrm{NP}$ has not been separated from constant-depth circuits consisting of solely $\mathrm{Mod}_6$ gates. So, the question is whether current techniques are enough to deduce interestingly ...
1answer
261 views

How does the problem of having a coffee-machine close relate to vertex cover?

Meeting rooms on university campuses may or may not contain coffee machines. We would like to ensure that every meeting room either has a coffee machine or is close enough to a meeting room that ...
1answer
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Examples of undecidable problems whose intersection is decidable

I know that given two problems are undecidable it does not follow that their intersection must be undecidable. For example, take a property of languages $P$ such that it is undecidable whether the ...
2answers
200 views

How does one find out whether $N = a^b$ for some $b$?

I was trying to find out how to find whether $N$ is a perfect power or not for some $a$ and $b$ (so the algorithm should discover that its not a perfect power if its not expressable in the form $a^b$)...
2answers
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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 ...
1answer
252 views

Feasibility of linear inequalities with binary variables

I have a system of linear inequalities of the form $A^t x \leq b$, where each of the $x_i$'s is a binary variable in $\{0, 1\}$. Are there any known fast and practical algorithms that can find a ...