Resource-constrained project scheduling: computing lower bounds by solving minimum cut problems

We present a novel approach to compute Lagrangian lower bounds on the objective function value of a wide class of resource-constrained project scheduling problems. The basis is a polynomial-time algorithm to solve the following scheduling problem: Given a set of activities with start-time dependent costs and temporal constraints in the form of time windows, find a feasible schedule of minimum total cost. In fact, we show that any instance of this problem can be solved by a minimum cut computation in a certain directed graph. We then discuss the performance of the proposed Lagrangian approach when applied to various types of resource-constrained project scheduling problems. An extensive computational study based on different established test beds in project scheduling shows that it can significantly improve upon the quality of other comparably fast computable lower bounds