Exercises 10 - CPSC 413 F18.pdf

Show that φ g evaluates to true if and only if

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Show that φ g evaluates to true if and only if exactly one of the three variables is set to true and the other two variables are set to false . 5. Balanced formulas We say a formula ψ is in 3-CNF if it is the logical and of clauses, each of which is the logical or of exactly three distinct literals. Here is an example of a formula in 3-CNF, ψ 1 = ( x 2 x 3 x 4 ) ( x 1 x 2 x 4 ) ( x 1 x 3 x 4 ) ( x 2 x 3 x 4 ) ( x 1 x 2 x 3 ) . We say an assignment to n variables x 1 , x 2 , . . . , x n is balanced if exactly n 2 of the variables are set to true and exactly n 2 of the variables are set to false . Given a 3-CNF formula φ , we want to find a balanced assignment that satisfies φ . 3SAT Input: A formula φ in which each clause has exactly 3 distinct literals Output: “Yes”, if there exists an assignment that satisfies φ . “No” otherwise. Balanced 3SAT Input: A formula φ in which each clause has exactly 3 distinct literals Output: “Yes”, if there exists a balanced assignment that satisfies φ . “No” otherwise. 1. Show that the formula ψ 1 above is a “Yes”-instance to Balanced 3SAT . 2. Show that 3SAT 6 m P Balanced 3SAT . Argue that your reduction f is correct, and argue briefly that your reduction runs in polynomial time.
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6. Monotone Satisfiability See Exercise 6 on page 507 in Chapter 8 in which the problem Monotone Satisfiability is being defined. Show that 3SAT 6 m P Monotone Satisfiability . 7. 2SAT can be solved in polynomial time (Harder) Give a polynomial time algorithm for 2SAT . ( Hint: This is a harder exercise. We want to relate 2SAT to a graph problem. Let φ be a 2SAT formula on n variables. Construct a graph G = ( V, E ) on 2 n vertices labeled by the 2 n literals. Consider how we can let the edges in E correspond to clauses of φ .) 8. Vertex Cover Our friend Merlin is coming to our assistance again, this time to help us solve the following vertex cover problem. We are given a graph G = ( V, E ) on n = | V | vertices, and we also given an integer k .
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  • Fall '13
  • GeoffCruttwell
  • Dynamic Programming, Quantification, vertex cover, independent set, NP-complete

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