Minimum ratio canceling is oracle polynomial for linear programming, but not strongly polynomial, even for networks

Authors:
S.Thomas Mccormick;Akiyoshi Shioura
Affiliations:
Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, BC, Canada V6T 1Z2;Department of Mechanical Engineering, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan
Venue:
Operations Research Letters
Year:
2000

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Abstract

This paper shows that the minimum ratio canceling algorithm of Wallacher (Unpublished manuscript, Institut fur Angewandte Mathematik, Technische Universitat, Braunschweig (1989)) (and a faster relaxed version) can be generalized to an algorithm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.