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.

L = {f in SAT | the number of satisfying assignments of f > 1/2 |f| } Show that L is NP-complete

4. Reducibility, NP-completeness 4a. Recall that linear ax + b = 0 is reducible to the quadratic ax²+bx+c=0. Show that the quadratic ax2+bx+c=0 is reducible to quartic ax*+bx+cx+dx+e=0 (Lec14.1). Is quartic reducible to quadratic? 4b. Consider the 4-way Partition Problem: Given a set S of positive integers, determine whether S can be partitioned into four subsets S1, S2, S3, and S4 such that the sum of the numbers in S, is equal to S2, S3, and S4. Demonstrate that the problem is NP-complete based on NP-completeness of the 2-way Partition Problem. 4c. Prove that P != NP.

Course: Linear Programming