The existence of a strongly polynomial time simplex algorithm for linear programs

TL;DR

This paper proves the existence of a strongly polynomial time simplex algorithm for linear programs, showing it can solve LPs with a number of steps bounded by the number of variables, using a novel parameter analysis technique.

Contribution

It introduces a new simplex algorithm with strongly polynomial complexity, addressing a long-standing open problem in optimization.

Findings

Existence of a strongly polynomial simplex algorithm proven

Number of pivot steps bounded by the number of variables

Uses a novel parameter analysis technique

Abstract

It is well known that the most challenging question in optimization and discrete geometry is whether there is a strongly polynomial time simplex algorithm for linear programs (LPs). This paper gives a positive answer to this question by using the parameter analysis technique presented by us (arXiv:2006.08104). We show that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem.