A compact mixed integer linear formulation for safe set problems

The Safe Set Problem is a type of partitioning problem with particular constraints on the weight of adjacent partition components. Several theoretical results exist in the literature along with a mixed integer linear formulation. We propose a new, more compact, Mixed Integer Linear Program for the (Weighted) Safe Set Problem and Connected Safe Set Problem with a polynomial number of variables and constraints, based on a flow from an additional fictitious node. The model is enriched by symmetry-breaking considerations and a variable reduction subproblem. We test the model on a benchmark of small instances which underline the difficulty of solving the Safe Set Problem through mathematical programming approaches. We also compare it with the only existing formulation in the literature and find that our approach is usually more efficient on a set of benchmark instances.