Minimum Ratio Canceling is Oracle Polynomial for Linear Programming, but Not Strongly Polynomial, Even for Networks

This paper shows that the minimum ratio canceling algorithm of Wallacher (1989) (and a faster relaxed version) can be generalized to an algorithm for general linear programs with geometric convergence. This implies that when we have a negative cycle oracle, this algorithm will compute an optimal solution in (weakly) polynomial time. We then specialize the algorithm to linear programming on unimodular linear spaces, and to the minimum cost flow and (dual) tension problems. We construct instances proving that even in the network special cases the algorithm is not strongly polynomial

Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis

International audienceIn this article, we propose a general framework for an algorithm derived from the primal simplex that guarantees a strict improvement in the objective after each iteration. Our approach relies on the identification of compatible variables that ensure a nondegenerate iteration if pivoted into the basis. The problem of finding a strict improvement in the objective function is proved to be equivalent to two smaller problems respectively focusing on compatible and incompatible variables. We then show that the improved primal simplex (IPS) of Elhallaoui et al. is a particular implementation of this generic theoretical framework. The resulting new description of IPS naturally emphasizes what should be considered as necessary adaptations of the framework versus specific implementation choices. This provides original insight into IPS that allows for the identification of weaknesses and potential alternative choices that would extend the efficiency of the method to a wider set of problems. We perform experimental tests on an extended collection of data sets including instances of Mittelmann's benchmark for linear programming. The results confirm the excellent potential of IPS and highlight some of its limits while showing a path toward an improved implementation of the generic algorithm