Com S 633: Randomness in Computation Lecture 11 Scribe: Dave Doty Summary: Com S 633: Randomness in Computation Lecture 11 Scribe: Dave Doty 1 Randomized Rounding Suppose we want to solve an optimization problem, such as MAX-SAT or MAX-CLIQUE. All most all optimization problems can always be converted into an instance of integer linear program (ILP). This means we have An instance of an ILP problem is a set of m+ n +mn constants (where m;n 2 Z + ) c 1 ; : : : ; c n ; d 1 ; : : : ; dm ; a 11 ; : : : ; amn 2 Q : The problem is to choose an assignment of values of x 1 ; : : : ; x n that minimize (or maximize) the objective function g(x 1 ; : : : ; x n ) = c 1 x 1 + c 2 x 2 + : : : + c n x n subject to the linear constraints a 11 x 1 + a 12 x 1 + : : : + a 1n x n  d 1 a 21 x 1 + a 22 x 1 + : : : + a 2n x n  d 2 : : : am1 x 1 + am2 x 1 + : : : + amnx n  dm and the additional constraint that each x i must be an integer. If this integer constraint is missing, then the problem is called linear programming (LP) and is polynomial-time solvable, whereas ILP is NP-hard. The rst polynomial-time algorithm for linear programming was the ellipsoid algorithm, due to Kachiyan. This was improved on by Karmaker, who invented Collections: Computer Technologies and Information Sciences