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# hw8 - CNF 2 is in P b Show that CNF 3 is NP-complete...

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CS 181 — Winter 2006 Problem Set #8 Formal Languages Due March 12, 2008 Problem 8.1. (10 points) Show that P is closed under union, concatenation, and star. Show that if P=NP, then given a graph G and integer k , you can construct in polynomial time a k -clique of G if one exists. (Careful: the NP problem only tells you whether some k -clique exists , it does not give you one!) Problem 8.2. (10 points) You can do this problem after we talk about NP-completeness. Let CNF k = {h φ i | φ is a satisﬁable CNF formula where each variable occurs at most k places } . a. Show that
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Unformatted text preview: CNF 2 is in P . b. Show that CNF 3 is NP-complete. Problem 8.3. (10 points) You can do this problem after we talk about NP-completeness on Monday. Let DBLSAT = {h φ i | φ has at least two satisfying assignments } . Show that DBLSAT is NP-complete. A subset of the nodes of a graph G is a dominating set if every other node of G is adjacent to some node in the subset. Let DOMSET = {h G , k i | G has a dominating set with k nodes } Show that DOMSET is NP-complete. For hardness, reduce from VERTEX-COVER ....
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