Constraint propagation techniques for the disjunctive scheduling problem

Constraint propagation is an elementary method for reducing the search space of combinatorial search and optimization problems which has become more and more important in the last decades. The basic idea of constraint propagation is to detect and remove inconsistent variable assignments that cannot participate in any feasible solution through the repeated analysis and evaluation of the variables, domains and constraints describing a specific problem instance. The contribution of this paper is twofold. The first contribution is a description of efficient constraint propagation methods also known as consistency tests for the disjunctive scheduling problem (DSP) which is a generalization of the classical job shop scheduling problem (JSP). Applying an elementary constraint based approach involving a limited number of search variables, we will derive consistency tests that ensure 3-b-consistency. We will further present and analyze both new and classical consistency tests which to some extent are generalizations of the aforementioned consistency tests involving a higher number of variables, but still can be implemented efficiently with a polynomial time complexity. Further, the concepts of energetic reasoning and shaving are analyzed and discussed. The other contribution is a classification of the consistency tests derived according to the domain reduction achieved. The particular strength of using consistency tests is based on their repeated application, so that the knowledge derived is propagated, i.e., reused for acquiring additional knowledge. The deduction of this knowledge can be described as the computation of a fixed point. Since this fixed point depends upon the order of the application of the tests, we first derive a necessary condition for its uniqueness. We then develop a concept of dominance which enables the comparison of different consistency tests as well as a simple method for proving dominance. An extensive comparison of all consistency tests is given. Quite surprisingly, we will find out that some apparently stronger consistency tests are subsumed by apparently weaker ones. At the same time an open question regarding the effectiveness of energetic reasoning is answered.