Polytope Extensions with Linear Diameters

TL;DR

This paper explores polytope extensions with linear diameters and links the efficiency of pivot rules in the simplex algorithm to the possibility of strongly polynomial-time solutions for linear programming.

Contribution

It introduces constructions of extended formulations that relate the diameter bounds to the complexity of solving linear programming problems.

Findings

Establishes a relaxed version of the Hirsch conjecture.

Shows that polynomial bounds on pivot rule steps imply strongly polynomial LP solutions.

Connects polytope diameter bounds to algorithmic complexity of linear programming.

Abstract

We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch conjecture and prove that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by a polynomial in the diameter plus the number of facets of the polyhedron of feasible solutions then the general linear programming problem can be solved in strongly polynomial time.