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Unformatted text preview: Fall Semester 2008 CS300 Algorithms Solution #10 1. The clique problem is in NP: Given an instance of the problem and a proposed vertices K , we can check in polynomial time whether K has k pairwise adjacent vertices. Let F = V 1 ≤ i ≤ p C i be a formula in CNF. We shall show how to construct from F a graph G such that G will have a clique of size k if and only if F is satisfiable. For any F , we define V and E as given in the hint. If F is satisfiable then there is a set of truth values for x i (1 ≤ i ≤ n ), such that each clause is true with this assignment. Thus, with this assignment there is at least one literal σ in each C i such that σ is true. Let S = { < σ,i >  σ is true in C i } be a set containing exactly one < σ,i > for each i . Then S forms a clique in G of size k . Similarly, if G has a clique K = ( V ,E ) of size k then let S = { < σ,i >  < σ,i > ∈ V } . Clearly,  S  = k . Furthermore, if S = { σ  < σ,i > ∈ S for some i } then S cannot contain both a literal δ and its complement δ as there is no edge connecting < δ,i > and < δ,j > in G . Hence by setting x i = true if x i ∈ S and...
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 Spring '08
 Unkown
 Algorithms, Computational complexity theory, NPcomplete problems, independent set, NPcomplete, Boolean satisfiability problem

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