Introduction to Algorithms
Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.5, Problem 8E
Program Plan Intro

To prove that half 3-CNF satisfiablity problem is NP complete.

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Show that the following problem belongs to NP class: we are given a set S of integer numbers and an integer number t. Does S have a subset such that sum of its elements is t? Note: Data Structures and Algorithm problem
(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard. (b) The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}} is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}} has exactly two solutions. Show that Double-SAT is NP-hard.
We are given a 3-CNF formula with n variables and m clauses, where m is even, in the half 3-CNF satisfiability issue. We want to know if there is a true variable assignment where precisely half of the clauses evaluate to 0 and exactly half of the clauses evaluate to 1. prove the NP-completeness of the partial 3-CNF satisfiability issue.
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