An Exact Method for Bisubmodular Function Maximization

Bisubmodularity--a natural generalization of submodularity--applies to set functions with two arguments and appears in a broad range of applications, including coupled sensor placement in infrastructure, coupled feature selection in machine learning, and drug-drug interaction detection in healthcare. In this paper, we study maximization problems with bisubmodular objective functions. We propose valid linear inequalities, namely the bisubmodular inequalities, for the hypograph of any bisubmodular function. We show that maximizing a bisubmodular function is equivalent to solving a mixed-integer linear program with exponentially many bisubmodular inequalities. Using this representation in a delayed constraint generation framework, we design the first exact algorithm to solve general bisubmodular maximization problems. Our computational experiments on the coupled sensor placement problem demonstrate the efficacy of our algorithm in constrained nonlinear bisubmodular maximization problems for which no existing exact methods are available.