Multivalued Decision Diagrams for Sequencing Problems

Abstract

Sequencing problems are among the most widely studied problems in operations research. Specific variations of sequencing problems include single machine scheduling, the traveling salesman problem with time windows, and precedence-constrained machine scheduling. In this work we propose a new approach for solving sequencing problems based on multivalued decision diagrams (MDDs). Decision diagrams are compact graphical representations of Boolean functions, originally introduced for applications in circuit design by Lee [7], and widely studied and applied in computer science. They have been recently used to represent the feasible set of discrete optimization problems, as demonstrated in [2] and [3, 4]. This is done by perceiving the constraints of a problem as a Boolean function f(x) representing whether a solution x is feasible. Nonetheless, such MDDs can grow exponentially large, which makes any practical computation prohibitive in general.

To circumvent this issue, Andersen et al. [1] introduced the concept of a relaxed MDD, which is a diagram of limited size that represents an over-approximation of the feasible solution set of a problem. We argue in this paper that such MDDs can be particularly useful as a discrete relaxation of the feasible set of sequencing problems. In particular, we embed relaxed MDDs within a stateof- the-art constraint-based scheduling system, and show that the resulting MDD propagation can reduce the solving time by several orders of magnitude.

This is a summary of the paper “A. A. Cire and W.-J. van Hoeve. Multivalued Decision Diagrams for Sequencing Problems. Operations Research, 61(6): 1411–1428, 2013”.