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CSIS 0250B Design and Analysis of Algorithms
Assignment 4
Due: 11:55 PM, Apr 16, 2009
1. Consider the CLIQUE problem restricted to graphs in which every vertex has degree at most
3. Call this problem CLIQUE3.
(a) Prove that CLIQUE3 is in NP.
(b) What is wrong with the following proof of NPcompleteness for CLIQUE3? We know
that the CLIQUE problem in general graphs is NPcomplete, so it is enough to present a
reduction from CLIQUE3 to CLIQUE. Given a graph G with vertices of degree at most 3,
and a parameter g, the reduction leaves the graph and the parameter unchanged: clearly the
output of the reduction is a possible input for the CLIQUE problem. Furthermore, the answer
to both problems is identical. This proves the correctness of the reduction and, therefore, the
NPcompleteness of CLIQUE3.
(c) It is true that the VERTEX COVER problem remains NPcomplete even when restricted
to graphs in which every vertex has degree at most 3. Call this problem VC3. What is
wrong with the following proof of NPcompleteness for CLIQUE3? We present a reduction
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This note was uploaded on 03/01/2010 for the course CS 1234 taught by Professor Chan during the Spring '10 term at University of the BíoBío.
 Spring '10
 Chan
 Algorithms

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