2 the Simplex Method

Linear programming has many important practical applications, and has also given rise to a wide body of theory. See Section 50.9 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 50.2 we consider the simplex method, in Section 50.3 we review deterministic linear time algorithms, in Section 50.4 randomized algorithms, and in Section 50.5 we consider the derandomization of the latter. Section 50.6 discusses the combinatorial framework of LP-type problems, which underlie most current combinatorial algorithms and allows their application to a host of optimization problems. We briefly describe the more recent combinatorial framework of unique sink orientations, in the context of striving for algorithms with a milder dependence on d. In Section 50.7 we examine parallel algorithms for this problem, and finally in Section 50.8 we briefly discuss related issues. The emphasis throughout is on complexity-theoretic bounds for the linear programming problem in the form (50.1.1).