Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications

Submodularity is an important concept in integer and combinatorial optimization. A classical submodular set function models the utility of selecting homogenous items from a single ground set, and such selections can be represented by binary variables. In practice, many problem contexts involve choosing heterogenous items from more than one ground set or selecting multiple copies of homogenous items, which call for extensions of submodularity. We refer to the optimization problems associated with such generalized notions of submodularity as Generalized Submodular Optimization (GSO). GSO is found in wide-ranging applications, including infrastructure design, healthcare, online marketing, and machine learning. Due to the often highly nonlinear (even non-convex and non-concave) objective function and the mixed-integer decision space, GSO is a broad subclass of challenging mixed-integer nonlinear programming problems. In this tutorial, we first provide an overview of classical submodularity. Then we introduce two subclasses of GSO, for which we present polyhedral theory for the mixed-integer set structures that arise from these problem classes. Our theoretical results lead to efficient and versatile exact solution methods that demonstrate their effectiveness in practical problems using real-world datasets

Maximizing k-submodular functions and beyond

We consider the maximization problem in the value oracle model of functions defined on $k$-tuples of sets that are submodular in every orthant and $r$-wise monotone, where $k\geq 2$ and $1\leq r\leq k$. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of $1/(1+r)$. For $r=k$, we give an analysis of a randomised greedy algorithm that shows that any such function can be approximated to a factor of $1/(1+\sqrt{k/2})$. In the case of $k=r=2$, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of $1/2$. We show that, as in the case of submodular functions, this result is the best possible in both the value query model, and under the assumption that $NP\neq RP$. Extending a result of Ando et al., we show that for any $k\geq 3$ submodularity in every orthant and pairwise monotonicity (i.e. $r=2$) precisely characterize $k$-submodular functions. Consequently, we obtain an approximation guarantee of $1/3$ (and thus independent of $k$) for the maximization problem of $k$-submodular functions