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Unformatted text preview: CS 154 Intro. to Automata and Complexity Theory Handout 25 Autumn 2006 David Dill November 14, 2006 Problem Set 6 Due: November 28, 2006 Homework: (Total 100 points) Do the following exercises. Problem 1. [20 points] Prove that the following problem, called 4TA-SAT, is NP-complete. The problem is defined as follows: INPUT: A boolean formula F ( X 1 , X 2 , . . . , X n ). PROBLEM: Does F have at least 4 satisfying truth assignments? You should reduce the 3-SAT problem to 4TA-SAT. The answers to the following questions constitute the proof of NP-completeness. (a). Prove that 4TA-SAT is in NP. (b). Describe a polynomial-time reduction from 3-SAT to 4TA-SAT. Your reduction should take a 3-SAT formula F and construct an instance of the 4TA-SAT problem, say the formula G . ( Hint : Suppose you add a variable Y to the boolean formula F , but dont actually use it in any of the clauses. If F had k satisfying truth assignments originally, how many satisfying truth assignments will it have now?) (c). Show that F is satisfiable if and only if G has at least four satisfying truth assignments. Problem 2. [20 points] Prove that the following problem called DS (Dominating Set) is NP- complete. A dominating set in a graph G = ( V, E ) is a set of vertices S...
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