ABSTRACT
Many combinatorial optimization problems can be cast as integer linear programming problems. A linear programming relaxation of an integer program provides a natural lower bound (in case of minimization problems) on the value of the optimal integral solution. An optimal solution to the linear programming relaxation may not necessarily be integral. This chapter illustrates the technique due to Raghavan and Thompson. They developed the first constant factor approximation algorithm for the minimum width routing problem in two dimensions. The idea of simultaneously rounding a set of variables, was used by Bertsimas et al. to establish the integrality of several well known polytopes. In particular they established the integrality of the polytopes associated with the minimum s-t cut, p-median on a cycle, uncapacitated lot sizing, and boolean optimization. Scaling is an important technique that has been applied to covering problems such as Vertex Cover to obtain a simple 2 factor approximation.
