A primal-simplex based Tardos' algorithm

TL;DR

This paper introduces a modified primal-simplex based algorithm for linear programming that is strongly polynomial under certain conditions, combining Tardos's original approach with recent simplex method enhancements.

Contribution

It presents a new primal-only algorithm for linear programming that leverages simplex and is strongly polynomial for totally unimodular matrices.

Findings

Algorithm is strongly polynomial for totally unimodular matrices.

Utilizes simplex method for auxiliary problems within Tardos's framework.

Provides a primal-only approach to Tardos' algorithm.

Abstract

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification and Mizuno's use of the simplex method, we introduce a modification of Tardos' algorithm considering only the primal problem and using simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate.