An enhanced primal-simplex based tardos' algorithm for linear optimization

Abstract

While the algorithmic complexity is in general worse than the one of Tardos’ original algorithms, the authors, motivated by the practicality of such methods, recently proposed a simplex-based variant that is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are nondegenerate. In this paper, we introduce a slight modification that circumvents the determination of the largest sub-determinant while keeping the same theoretical performance. Assuming that the coefficient matrix is integer-valued and the auxiliary problems are non-degenerate, the proposed algorithm is polynomial in the dimension of the input data and the largest absolute value of a sub-determinant of the coefficient matrix.