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Chapter 7: Q5E (page 322)
Is the following formula satisfiable?
This technique satisfiable since it uses every one of the variables that produce true and pair values based on X and Y.
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Let collection of subsets of S, for some , such that elements of S can be colored red or blue so that no Ci has all its elements colored with the same color}. Show that is NP-complete.
Let be the subset sum problem in which all numbers are represented in unary. Why does the completeness proof for fail to show is complete? Show that
For a cnf-formula with variables and clauses , show that you can construct in polynomial time an NFA with states that accept all nonsatisfying assignments, represented as Boolean strings of length . Conclude that implies that NFAs cannot be minimized in polynomial time.
A cut in an undirected graph is a separation of the vertices V into two disjoint subsets S and T . The size of a cut is the number of edges that have one endpoint in S and the other in T . Let
Show that MAX-CUT is NP-complete. You may assume the result of Problem 7.26. (Hint: Show that. The variable gadget for variable x is a collection of 3c nodes labeled with x and another nodes labeled with x . The clause gadget is a triangle of three edges connecting three nodes labeled with the literals appearing in the clause. Do not use the same node in more than one clause gadget. Prove that this reduction works.)
The difference hierarchyis defined recursively as
(Here .) For example, a language in D2P is the difference of two NP languages. Sometimes is called DP (and may be written DP). Let
.Show that Z is complete for DP. In other words, show that Z is in DP and every language in DP is polynomial time reducible to Z.
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