# Submodular maximization and its generalization through an intersection cut lens

## TL;DR

This paper introduces a novel intersection cut approach for submodular maximization problems, utilizing a convex extension and a hybrid algorithm, leading to improved solutions for various combinatorial optimization tasks.

## Contribution

It develops a new intersection cut framework based on a convex extension of submodular functions, with an efficient hybrid algorithm, and extends the method to nonconvex models like submodular-supermodular functions.

## Key findings

- Effective intersection cuts improve MIP solutions for submodular problems.
- The hybrid discrete Newton algorithm computes cuts efficiently and exactly.
- Applications include max cut, pseudo Boolean maximization, and Bayesian D-optimal design.

## Abstract

We study a mixed-integer set $S:=\{(x,t) \in \{0,1\}^n \times \mathbb{R}: f(x) \ge t\}$ 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 $S$. We construct a continuous extension $F$ of $f$, which is convex and defined over the entire space $\mathbb{R}^n$. We show that the epigraph of $F$ is an $S$-free set, and characterize maximal $S$-free sets including the epigraph. 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, 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.