Min cost flow on unit capacity networks and convex cost K-flow are as easy as the assignment problem with All-Min-Cuts algorithm

Abstract

We explore here surprising links between the time-cost-tradeoff problem and the minimum cost flow problem that lead to fast, strongly polynomial, algorithms for both problems. One of the main results is a new algorithm for the unit capacity min cost flow that matches the complexity of the fastest strongly polynomial algorithm known for the assignment problem. The time cost tradeoff (TCT) problem in project management is to expedite the durations of activities, subject to precedence constraints, in order to achieve a target project completion time at minimum expediting costs, or, to maximize the net benefit from a reward associated with project completion time reduction. Each activity is associated with integer normal duration, minimum duration, and expediting cost per unit reduction in duration. We devise here the {\em all-min-cuts} procedure, which for a given maximum flow, is capable of generating all minimum cuts (equivalent to minimum cost expediting) of equal value very efficiently. Equivalently, the procedure identifies all solutions that reside on the TCT curve between consecutive breakpoints in average $O(m+n \log n)$ time, where $m$ and $n$ are the numbers of arcs and nodes in the network. The all-min-cuts procedure implies faster algorithms for certain TCT problems and for certain min cost flow problem with a significantly different approach from the ones known.