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Unformatted text preview: Fall Semester 2008 CS300 Algorithms Homework #10 (due 12/12) 1. The clique problem is defined as follows: Given a graph G and a positive integer k , does G have k pairwise adjacent vertices? (A set of k pairwise adjacent vertices is called a kclique, i.e., an induced subgraph which is a complete graph K n .) Show that the clique problem is NPcomplete. Hint. Use the following polynomial transformation to reduce the satisfiability problem to the clique problem. Suppose that C 1 ,C 2 , ··· C p are the clauses in a CNF expression and let the variables in the expression be denoted by x i , 1 ≤ i ≤ n . We refer to either a variable or its negation as a literal. The expression is transformed to the graph with V = { < σ,i >  σ is a literal in clause C i } , i.e., V has a vertex representing each occurrence of a literal in a clause, and E = { ( < σ,i >,< δ,j > )  i 6 = j and σ 6 = δ } ....
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 Spring '08
 Unkown
 Algorithms, vertex cover, NPcomplete problems, independent set, Boolean satisfiability problem, pairwise adjacent vertices

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