38 LINEAR PROGRAMMING

Linear programming has many important practical applications, and has also given rise to a wide body of theory. See Section 38.9 for recommended sources. Here we consider the linear programming problem in the form of maximizing a linear function of d variables subject to n linear inequalities. We focus on the relationship of the problem to computational geometry, i.e., we consider the problem in small dimension. More precisely, we concentrate on the case where d ≪ n, i.e., d = d(n) is a function that grows very slowly with n. By linear programming duality, this also includes the case n ≪ d. This has been called fixed-dimensional linear programming, though our viewpoint here will not treat d as constant. In this case there are strongly polynomial algorithms, provided the rate of growth of d with n is small enough. The plan of the chapter is as follows. In Section 38.2 we consider the simplex method, in Section 38.3 we review deterministic linear time algorithms, in Section 38.4 randomized algorithms, and in Section 38.5 we consider the derandomization of the latter. Section 38.6 discusses combinatorial framework of LP-type problems which is underlying most current combinatorial algorithms and allows their application to a host of optimization problems. In Section 38.7 we examine parallel algorithms, and finally in Section 38.8 we briefly discuss related issues. The emphasis throughout is on complexity-theoretic bounds for the linear programming problem in the form 38.1.1.