Log out Go to App
Go to App

Learning Materials

Features

Discover

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 and only if it reduces to an NP-Complete problem. This is due to the definitions of NP-Easy and NP-Complete problems, in which NP-Easy problems can be reduced to with any problem in NP while NP-Complete is a class of problems that every problem in NP can be reduced to.

Step by step solution

01

Definition of NP-Easy

Understand that NP-Easy is a class of problems to which all problems in NP can be reduced. In other words, if there exists a polynomial time reduction from any problem in NP to a problem X, then X is NP-Easy.
02

Definition of NP-Complete

Understand that NP-Complete refers to the hardest problems in NP, in the sense that they are the ones to which any problem in NP can be reduced in polynomial time. It means, a problem Y is NP-Complete if it is in NP, and all problems in NP can be reduced to Y in polynomial time.
03

Analyzing the 'if' Part

When a problem Z reduces to an NP-Complete problem, it implies that there is a polynomial time algorithm that transforms instances of problem Z into instances of an NP-Complete problem. Thus, Z can be solved utilizing the solutions to the NP-Complete problem, which signifies that Z is in NP-Easy.
04

Analyzing the 'only if' Part

For a problem Z to be in NP-Easy, there must be a way to reduce any problem in NP to it in polynomial time. Given that NP-Complete problems are in NP, Z must be able to reduce to any NP-Complete problem. Hence, if a problem is NP-easy, it reduces to an NP-Complete problem.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given a list of \(n\) positive integers \((n \text { even }),\) divide the list into two sublists such that the difference between the sums of the integers in the two sublists is minimized. Is this problem an \(N P\) complete problem? Is this problem an \(N P\) -hard problem?

Write a polynomial-time algorithm that checks if an undirected graph has a Hamiltonian Circuit, assuming that the graph has no vertex with degree exceeding 2

For the Sum-of-Subsets problem discussed in Chapter \(5,\) can you develop an approximation algorithm that runs in polynomial time?

Show that the reduction of the Hamiltonian Circuits Decision problem to the Traveling Salesperson (Undirected) Decision problem can be done in polynomial time.

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.
See all solutions

Recommended explanations on Computer Science Textbooks

Algorithms in Computer Science

Read Explanation

Computer Systems

Read Explanation

Game Design in Computer Science

Read Explanation

Computer Organisation and Architecture

Read Explanation

Theory of Computation

Read Explanation

Data Structures

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free