CSC 2411 - Linear Programming and Combinatorial Optimization ∗ Lecture 11 : Primal - Dual Schema and Algorithms

This lecture starts with an LP relaxation of the max sat problem and a proof showing how close the relaxed optimum is to the integral optimum. Then we move on to a definition and proof complementary slackness and use it to establish primal-dual schema and algorithms. We then show an example of using such an algorithm on the set cover problem. The maximum satisfiability problem asks for the maximum number of conjunctive normal form (CNF) clauses that can be satisfied by any assignment of the variables involved: Input: CNF formula (e.g. (x 1 ∨ ¬x 2 ∨ x 3) ∧ (x 2 ∨ x 1) ∧ (¬x 3 ∨ ¬x 1)) Output: Truth assignments satisfying maximum number of clauses(e.g. from above: x 1 = 1; x 2 = 1; x 3 = 0)