Abstract
Submodularity defines a general framework rich of important theoretical properties while accommodating numerous applications. Although the notion has been present in the literature of Combinatorial Optimization for several decades, it has been overlooked in the analysis of global constraints. The current work illustrates the potential of submodularity as a powerful tool for such an analysis. In particular, we show that the cumulative constraint, when all tasks are identical, has the submodular/supermodular representation property, i.e., it can be represented by a submodular/supermodular system of linear inequalities. Motivated by that representation, we show that the system of any two (global) constraints not necessarily of the same type, each bearing the above-mentioned property, has an integral relaxation given by the conjunction of the linear inequalities representing each individual constraint. This result is obtained through the use of the celebrated polymatroid intersection theorem.
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