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

TL;DR

This paper introduces a new, efficient algorithm for the minimum cost flow problem on unit capacity networks, matching the complexity of the best algorithms for the assignment problem, and links it to the time-cost tradeoff problem.

Contribution

The paper presents the All-Min-Cuts algorithm, which efficiently finds all minimum cuts and solutions on the TCT curve, leading to faster algorithms for specific flow and project management problems.

Findings

New strongly polynomial algorithm for unit capacity min cost flow.

Efficient generation of all minimum cuts for given flow.

Faster algorithms for time-cost tradeoff problems in project management.

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.