hbs12 - Problem 2 . [Self-Reductions] In each case below,...

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CS 473: Algorithms, Fall 2010 HBS 12 Problem 1 . [Graph Isomorphism] Two graphs are said to be isomorphic if one can be transformed into the other just by relabeling the vertices. For example, the graphs shown below are isomorphic; the left graph can be transformed into the right graph by the relabeling (1 , 2 , 3 , 4 , 5 , 6 , 7) = ( c,g,b,e,a,f,d ). Consider the following related decision problems: GraphIsomorphism: Given two graphs G and H , determine whether G and H are isomorphic. EvenGraphIsomorphism: Given two graphs G and H , such that every vertex in G and H has even degree, determine whether G and H are isomorphic. SubgraphIsomorphism: Given two graphs G and H , determine whether G is isomorphic to a subgraph of H . a.) Describe a polynomial-time reduction from GraphIsomorphism to EvenGraphIsomorphism. b.) Describe a polynomial-time reduction from GraphIsomorphism to SubgraphIsomorphism. c.) Prove that SubgraphIsomorphism is NP-complete by reducing from Clique.
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Unformatted text preview: Problem 2 . [Self-Reductions] In each case below, assume that you are given a black box which can answer the decision version of the indicated problem. Use a polynomial number of calls to the black box to construct the desired set. Independent set: Given a graph G and an integer k , does G have a subset of k vertices that are pairwise nonadjacent? Subset sum: Given a multiset (elements can appear more than once) X = x 1 ,...,x k of positive integers, and a positive integer S , does there exist a subset of X with sum exactly S ? k-Color: Given a graph G , is there a proper k-coloring? In other words, can we assign one of the k colors to each node such that no node is adjacent to a node of the same color? 1...
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