Some Experiments with Submodular Function Maximization via Integer Programming

Submodular function maximization is a classic problem in optimization, with many real world applications, like sensor coverage, location problems and feature selection, among others. Back in the 80’s, Nemhauser and Wolsey proposed an integer programming formulation for the general submodular function maximization. Being the number of constraints in the formulation exponential in the size of the ground set, a constraint generation technique was proposed. Since then, the method was not developed further. Given the renewed interest in recent years in submodular function maximization, the constraint generation method has been used as reference to evaluate both exact and heuristic approaches. However, the outcome of those experiments was that the method is utterly slow in practice, even for small instances. In this paper we propose several algorithmic enhancements to the constraint generation method. Preliminary computational results show that a proper implementation, while still not scalable to big instances, can be significantly faster than the obvious implementation by the book. A comparison with direct mixed integer linear programming formulations on some classes of models that admit one also show that the submodular framework, in its generality, is clearly slower than dedicated formulations, so it should be used only when those approaches are not viable.