Near-Equivalence of Network Flow Algorithms.

Abstract : Network problems arising in practice typically have non-negative arc costs. On such problems we show that the following algorithms perform, modulo ties, the same sequence of flow augmentations. (1) Simplex (with the standard pivot rule and Big-M start), (2) Out-of-Kilter (Primal-Dual), (3) Dual Simplex (with the standard pivot rule), (4) Lemke's Complementary Pivot Algorithm. All methods compute a shortest path tree by mimicking the Dijkstra algorithm and then send flow along a sequence of minimum cost paths. Differences in implementation are discussed. It becomes clear that Dantzig's simplex method with the best empirical pivot rule (not the standard rule) will outperform other methods (variations of Simplex with the standard rule). A simple reason is given why Dual Simplex (best empirical) cannot do as well as Simplex (best empirical).