Set Problem is in NP.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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3. The Independent Set Problem is:
Given a graph G and an integer K, is there a set S of K vertices in G such that no two vertices
in S are adjacent?
The Exactly Half Independent Set Problem is:
Given a graph G on n vertices where n is even, is there a set S of n/2 vertices in G such that
no two vertices in S are adjacent?
(a) Prove that the Exactly Half Independent Set Problem is in NP.
(b) Prove that the Independent Set Problem reduces to the Exactly Half Independent Set Problem in
polynomial time. (Hint: Given a graph G in the Independent Set Problem, add vertices and edges
to turn the problem into the Exactly Half Independent Set Problem. For your reduction, you will
need to consider separately the cases of independent problems where n > 2K from the cases where
n<2K.)
(Note that proving a) and b) proves that the Exactly Half Independent Set Problem is NP-complete.)
Transcribed Image Text:3. The Independent Set Problem is: Given a graph G and an integer K, is there a set S of K vertices in G such that no two vertices in S are adjacent? The Exactly Half Independent Set Problem is: Given a graph G on n vertices where n is even, is there a set S of n/2 vertices in G such that no two vertices in S are adjacent? (a) Prove that the Exactly Half Independent Set Problem is in NP. (b) Prove that the Independent Set Problem reduces to the Exactly Half Independent Set Problem in polynomial time. (Hint: Given a graph G in the Independent Set Problem, add vertices and edges to turn the problem into the Exactly Half Independent Set Problem. For your reduction, you will need to consider separately the cases of independent problems where n > 2K from the cases where n<2K.) (Note that proving a) and b) proves that the Exactly Half Independent Set Problem is NP-complete.)
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