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Unformatted text preview: CS 154 Intro. to Automata and Complexity Theory Handout 21 Autumn 2009 David Dill November 3, 2009 Problem Set 5 Due: November 10, 2008 Homework: (Total 100 points) Do the following exercises. Problem 1. [20 points] Prove that the following problem, called 4TASAT, is NPcomplete. 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 3SAT problem to 4TASAT. The answers to the following questions constitute the proof of NPcompleteness. (a). Prove that 4TASAT is in NP. (b). Describe a polynomialtime reduction from 3SAT to 4TASAT. Your reduction should take a 3SAT formula F and construct an instance of the 4TASAT problem, say the formula G . ( Hint : Suppose you add a variable Y to the boolean formula F , but don’t 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...
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 '08
 Motwani,R
 NPcomplete, Boolean formula, David Dill

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