By by Vilhelm Dahllöf.
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Additional info for Exact algorithms for exact satisfiability problems
For four clauses x = (a ∨ b ∨ c ∨ d ∨ e), y = (a ∨ b ∨ f ∨ g ∨ h), z = (a ∨ C1 ) and w = (b ∨ C2 ) such that some of c – h are heavy or appears in a 4-clause w = (α ∨ C3 ) such that α is heavy; return DX(F ∧ (a ∨ b)) or DX(F (a/f alse; b/f alse)) 10. For two clauses x = (a ∨ b ∨ c ∨ d ∨ e), y = (a ∨ b ∨ f ∨ g) such that some of c – e are heavy or appears in a 4-clause w = (α ∨ C3 ) such that α is heavy; return DX(F ∧ (a ∨ b))) or DX(F (a/f alse; b/f alse)) 11. For two clauses x = (a ∨ b ∨ c ∨ d) and y = (b ∨ e ∨ f ∨ g) such that a is heavy and no member of x or y is a singleton; return DX(F ∧ (b ∨ e)) or DX(F (b/f alse; e/f alse)) 12.
There are no three clauses x = (a∨b∨c∨e), y = (a∨b∨C1 ) and z = (a ∨ c ∨ C2 ) such that δ(a) = 3, e is a singleton and δ(c) = δ(b) = 2 If the above configuration exists, then replace x, y and z by (b ∨ C1 ) and (c ∨ C2 ) 2. Preliminaries 39 Lemma 1. The process of applying the above transformations on a formula F terminates in polynomial time, measuring in n(F ). Proof. All transformations but the first two run in polynomial time and are guaranteed to remove variables. When transformation 2 does not remove any variables it takes constant time (remember that when measuring in n we disregard any polynomial factors in the length of the formula) and the same holds for transformation 1.
E. the corresponding Sat problem. Given the lack of results for Max sat, we get an indication of the hardness of Max xsat in terms of upper time bounds. We will make more comparisons between Max xsat and Max sat in Chapter 7. 6 Max Hamming Exact Satisfiability Most previous algorithms for optimisation problems have contented themselves with producing one best or good-enough solution. However, there is often an actual need for several solutions that are as different as possible. The notion of “as different as possible” can be captured using the concept of Hamming distance.
Exact algorithms for exact satisfiability problems by by Vilhelm Dahllöf.