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Unformatted text preview: Homework #7 (Solutions) 1. Consider the class of 3SAT 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 Hall’s Theorem is a statement about when a bipartite matching always occurs; this hint leads us to question what is being matched ? We can’t match literals to { true, false } , because we have more than two variables. In normal 3SAT, we don’t 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 Hall’s Theorem....
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 '06
 Shamsian
 Algorithms, Graph Theory, Big O notation, Analysis of algorithms, Shortest path problem, optimal ﬂow paths

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