Submodular maximization and its generalization through an intersection cut lens

Abstract

We study a mixed-integer set \(\mathcal {S}:=\{(x,t) \in \{0,1\}^n \times \mathbb {R}: f(x) \ge t\}\) arising in the submodular maximization problem, where f is a submodular function defined over \(\{0,1\}^n\). We use intersection cuts to tighten a polyhedral outer approximation of \(\mathcal {S}\). We construct a continuous extension \(\bar{\textsf{F}}_f\) of f, which is convex and defined over the entire space \(\mathbb {R}^n\). We show that the epigraph \({{\,\textrm{epi}\,}}(\bar{\textsf{F}}_f)\) of \(\bar{\textsf{F}}_f\) is an \(\mathcal {S}\)-free set, and characterize maximal \(\mathcal {S}\)-free sets containing \({{\,\textrm{epi}\,}}(\bar{\textsf{F}}_f)\). We propose a hybrid discrete Newton algorithm to compute an intersection cut efficiently and exactly. Our results are generalized to the hypograph or the superlevel set of a submodular-supermodular function over the Boolean hypercube, which is a model for discrete nonconvexity. A consequence of these results is intersection cuts for Boolean multilinear constraints. We evaluate our techniques on max cut, pseudo Boolean maximization, and Bayesian D-optimal design problems within a MIP solver.