HW7 solutions

HW7 solutions - Homework #7 (Solutions) 1. Consider the...

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Unformatted text preview: Homework #7 (Solutions) 1. Consider the class of 3-SAT instances in which each of the n variables occur counting posi- tive and negative appearances combined in exactly three clauses. Furthermore, no variable will show up twice in the same clause. Show how to find a satisfying assignment using network flow. Hint: How many clauses are there? Solution Halls Theorem is a statement about when a bipartite matching always occurs; this hint leads us to question what is being matched ? We cant match literals to { true, false } , because we have more than two variables. In normal 3-SAT, we dont know how many clauses we have, but perhaps we do here. Each of the literals occurs in exactly three clauses, for a total of 3 n appearances. Furthermore, each clause has 3 variables in it, for a total of n clauses. We can then set up a bipartite matching, with n variables and n clauses. There is an edge with capacity one from X i to C j if and only if X i appears in C j . We know from Halls Theorem....
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HW7 solutions - Homework #7 (Solutions) 1. Consider the...

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