New results on the average behavior of simplex algorithms

It has been a challenge for mathematicians to theoreti- cally confirm the extremely good performance of simplex algorithms for linear programming. We have confirmed that a certain variant of the simplex method solves problems of order m x n in an expected num- ber of steps which is bounded between two quadratic functions of the smaller dimension of the problem. Our probabilistic assumptions are rather weak. 1. Introduction. We consider the linear programming problem of order m x n in the form: Maximize cTx over all x >_ 0 in Rn such that Ax 5 6, where A E Rmxn. It has been observed that the simplex algorithms for linear programming, developed by George Dantzig (Dl, work extremely well. A central question in the field of analysis of algorithms is to estimate the expected number of steps that these algorithms perform relative to different probability distributions of inputs. Some background is given in 52. In (AM1 and T) we consider a variant of the simplex method called the "lexicographic self-dual method". We find that the expected number of steps of this variant, relative to a rather weak probabilistic model, is only O((min(m, n))2). This is the first polynomial average-case upper bound for a simplex algorithm which is capable of solving any linear programming problem. The first two authors (AM21 have also determined a quadratic lower bound, so the behavior of this variant is indeed quadratic, whereas it has been conjectured that other variants take a linear expected number of steps. With various choices of the starting point, this variant can simulate different "constraint-by-constraint" and "variable dimension" algorithms, as well as combinations thereof (Me2). Adler, Karp and Shamir (AKS) provide a different proof of similar upper bound for a certain class of "constraint-by-constraint" algorithms. It follows . from (Me21 that their variant can also be simulated by the self-dual algorithm with an appropriate starting point.