On the mixed set covering, packing and partitioning polytope

We study the polyhedral structure of the mixed set covering, packing and partitioning problem, which has drawn little attention in the literature but has many real-life applications. By considering the “interactions” between the different types of edges of an induced graph, we develop new classes of valid inequalities. In particular, we derive the (generalized) mixed odd hole inequalities, and identify sufficient conditions for them to be facet-defining. In the special case when the induced graph is a mixed odd hole, the inclusion of this new facet-defining inequality provide a complete polyhedral characterization of the mixed odd hole polytope. Computational experiments indicate that these new valid inequalities may be effective in reducing the computation time in solving mixed covering and packing problems.