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Chapter 9: Problem 14

Show that a problem is \(N P\) -easy if and only if it reduces to an \(N P\) -complete problem.

Short Answer

Expert verified
A problem is NP-Easy if it reduces to an NP-complete problem and vice-versa. This is because by definition, an NP-easy problem can be reduced to any problem in NP including NP-Complete problems. Feasibly, if a problem is reducible to an NP-Complete problem, it is essentially an NP-Easy problem due to the fact that it can be deduced in polynomial time.

Step by step solution

01

Understanding of NP-Easy and NP-Completeness

Start by understanding the terms. NP-Easy are problems that can be reduced to any problem in NP in polynomial time. NP-Complete problems are those that are in NP and any problem in NP can be reduced to them in polynomial time. Therefore, for a problem to be NP-Easy it means it can be reduced to an NP-complete problem.
02

Reduction to NP-Complete implies NP-Easiness

If a problem reduces to an NP-Complete problem in polynomial time, then by definition of NP-Easy, it is an NP-Easy.
03

NP-Easiness implies Reduction to an NP-Complete Problem

Conversely, if a problem is NP-Easy then by definition it can be reduced to any problem in the NP in polynomial time. Since NP-Complete problems are in NP, then the problem can be reduced to an NP-Complete problem. This shows the equivalence.

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Most popular questions from this chapter

When all instances of the CNF-Satisfiability Problem have exactly three literals per clause, it is called the 3 -Satisfiability Problem. Knowing that the 3-Satisfiability Problem is \(N P\) -complete, show that the Graph 3 -Coloring Problem is also \(N P\) -complete.

Show that a graph problem using the number of vertices as the measure of the size of an instance is polynomially equivalent to one using the number of edges as the measure of the size of an instance.

Can you develop an approximation algorithm for the CNF-Satisfiability Problem by stating it as an optimization problem- that is, by finding a truth assignment of the literals in the expression that makes the maximum possible number of clauses true?

Is the Towers of Hanoi Problem an NP-complete problem? Is it an \(N P\) -easy problem? Is it an \(N P\) -hard problem? Is it an \(N P\) -equivalent problem? Justify your answers. This problem is presented in Exercise 17 in Chapter 2.

Show that if a problem is not in \(N P\), it is not \(N P\) -easy. Therefore, Presburger Arithmetic and the Halting Problem are not \(N P\) -easy.
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