 
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 MAXSAT or MAXCLIQUE.
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 polynomialtime solvable,
whereas ILP is NPhard. The rst polynomialtime algorithm for linear programming was
the ellipsoid algorithm, due to Kachiyan. This was improved on by Karmaker, who invented
