10 - Dealing with hard problems - NPC (Solution)

10 - Dealing with hard problems - NPC (Solution) - The Hong...

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Unformatted text preview: The Hong Kong University of Science & Technology COMP 271: Design and Analysis of Algorithms Fall 2007 Tutorial 10: NP-Completeness Below is the solution to the tutorial questions. Question 1 : Given an undirected graph G = ( V, E ), a feedback vertex set is a subset of vertices such that every simple cycle of G passes through one of these vertices. The feedback vertex set problem (FVS) is, given a graph G and an integer k , does G contain a feedback vertex set of size at most k ? For example, the graph shown in the figure has a feedback vertex set of size 2 (shaded). Show that FVS is in NP. That is, given a graph G that has a FVS of size k , (a) Give a certificate for the FVS problem. Note: Certificate is a specific object which allows us to verify the given input (i.e. graph G ) is actually a yes-input. (b) Show how to use this certificate to verify the presence of a FVS. (Hint: Concentrate on the vertices that are not part of the FVS.) (c) Show the verification algorithm runs in polynomial time. Solution: (a) The certificate is the set V ′ of vertices that are in the feedback vertex set of size k . 1 (b) The verification algorithm needs to test whether every cycle in G passes through at least one of the vertices in V ′ . But, there are ex- ponentially many cycle in a graph! So, by testing each cycle would NOT run in polynomial time. Therefore, instead we can do something as follows. A new graph G ′ is constructed by deleting all the vertices in V ′ from G , along with any incident edges. Then, we notice that if every cycle in G passes through at least one of the vertices V ′ , there should have no cycle in G ′ anymore! The problem now is how to check whether there is any cycle in G ′ ! Well, we can do the checking by running DFS and see whether the resulting graph has AT LEAST ONE BACK EDGE or not. If so, V ′ is not a feedback vertex set (Since the cycle in G ′ does not pass through any vertex of V ′ ) and otherwise, it is! (c) Deleting all the vertices in V ′ from G along with any incident edges and running DFS to check whether there is any back edge can be performed in polynomial time....
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This note was uploaded on 10/18/2009 for the course COMP 271 taught by Professor Arya during the Spring '07 term at HKUST.

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10 - Dealing with hard problems - NPC (Solution) - The Hong...

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