For decades, researchers have complained that contiguity constraints make problems like districting much harder than other combinatorial optimization problems. This has prompted the use of integer programming models with inexact contiguity constraints that may be fast in practice but are invalid in the sense that they cut off or “overlook” some contiguous solutions. Examples include … Read more
We consider a fundamental question in political districting: How many counties can be kept whole (i.e., not split across multiple districts), while satisfying basic criteria like contiguity and population balance? To answer this question, we propose integer programming techniques based on combinatorial Benders decomposition. The main problem decides which counties to keep whole, and the … Read more
The parallel processor scheduling and bin packing problems with contiguity constraints are important in the field of combinatorial optimization because both problems can be used as components of effective exact decomposition approaches for several two-dimensional packing problems. In this study, we provide an extensive review of existing mathematical formulations for the two problems, together with … Read more