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

We explore here surprising links between the time-cost-tradeoff problem and the minimum cost flow problem that lead to faster, strongly polynomial, algorithms for both problems. One of the main results is a new algorithm for the unit capacity min cost flow that culminates decades of efforts to match 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 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 TCT problems that have “small” number of breakpoints in the respective TCT curve: For a project network on n nodes and m arcs, with n′ arcs of finite uniform expediting costs, the run time is O((n + n′)(m + n log n)); for projects with rewards of O(K) per unit reduction in the project completion time the run time is O((n+K)(m+ n log n)). Using the primal-dual relationship between TCT and the minimum cost flow problem (MCF) we generate faster strongly polynomial algorithms for various cases of minimum cost flow: For MCF on unit (vertex) capacity network, the complexity of our algorithm is O(n(m + n log n)), which is faster than any other strongly polynomial known to date; For a minimum convex (or linear) cost K-flow problem our algorithm runs in O((n + K)(m + n log n)); for MCF on n′ constant capacities arcs, the all-min-cuts algorithm runs in O((n+ n′)(m+ n log n)) steps. This complexity of the algorithm for any min cost O(n)-flow matches the best known complexity for the assignment problem, O(n(m + n log n)), yet with a significantly different approach. Our new methodology for general MCF problems departs dramatically from any methods currently in use, and as such it affords new and fresh insights in this well studied area, with a potential for numerous other improvements in complexities, beyond the ones reported here.