Q1a Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k. 1b INDEPENDENT SET has the following Decision Problem: given a graph G and an integer k₁, does G have an independent set of size at least k₁? VERTEX COVER has the following Decision Problem: given a graph G and an integer k2, does G have a vertex cover of size at most k₂? It is known that INDEPENDENT SET is NP-complete. Prove that INDEPENDENT SET SP VERTEX COVER (i.e that independent set is polynomial-time reducible to vertex cover). Use this to conclude that VERTEX COVER must also be NP-complete.

Q1a Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k. 1b INDEPENDENT SET has the following Decision Problem: given a graph G and an integer k₁, does G have an independent set of size at least k₁? VERTEX COVER has the following Decision Problem: given a graph G and an integer k2, does G have a vertex cover of size at most k₂? It is known that INDEPENDENT SET is NP-complete. Prove that INDEPENDENT SET SP VERTEX COVER (i.e that independent set is polynomial-time reducible to vertex cover). Use this to conclude that VERTEX COVER must also be NP-complete.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Q1a
Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly
explain why the minimum size of a vertex cover in G is n - k.
1b
INDEPENDENT SET has the following Decision Problem: given a graph G and an integer k₁,
does G have an independent set of size at least k₁?
VERTEX COVER has the following Decision Problem: given a graph G and an integer k2,
does G have a vertex cover of size at most k2?
It is known that INDEPENDENT SET is NP-complete. Prove that INDEPENDENT SET <p
VERTEX COVER (i.e that independent set is polynomial-time reducible to vertex cover). Use this
to conclude that VERTEX COVER must also be NP-complete.

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