On combinatorial network flows algorithms and circuit augmentation for pseudoflows

Abstract

There are numerous combinatorial algorithms for classical min-cost flow problems and their simpler variants like max flow or shortest path problems. It is well-known that many of these algorithms are related to the Simplex method and the more general circuit augmentation schemes: prime examples are the network Simplex method, a refinement of the primal Simplex method, and min-mean cycle canceling, which corresponds to a steepest-descent circuit augmentation scheme. We are interested in a deeper understanding of the relationship between circuit augmentation and combinatorial network flows algorithms. To this end, we generalize from primal flows to so-called pseudoflows, which adhere to arc capacities but allow for a violation of flow balance. We introduce ‘pseudoflow polyhedra,’ wherein slack variables are used to quantify this violation, and characterize their circuits. This enables the study of combinatorial network flows algorithms in view of the walks they trace in these polyhedra, and the pivot rules for the steps. In doing so, we provide an ‘umbrella,’ a general framework, that captures several algorithms. We show that the Successive Shortest Path Algorithm for min-cost flow problems, the Shortest Augmenting Path Algorithm for max flow problems, and the Preflow-Push algorithm for max flow problems lead to (non-edge) circuit walks in these polyhedra. The former two are replicated by circuit augmentation schemes for simple pivot rules. Further, we show that the Hungarian Method leads to an edge walk and is replicated, equivalently, as a circuit augmentation scheme or a primal Simplex run for a simple pivot rule.