Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.)
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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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Step 1: Half Independent Set Problem is in NP.
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Step 2: reduces to the Exactly Half Independent Set Problem in polynomial time.
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