An exact cutting plane method for k-submodular function maximization

A natural and important generalization of submodularity – $k$-submodularity – applies to set functions with $k$ arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In this paper, we study maximization problems with $k$-submodular objective functions. We propose valid linear inequalities, namely the $k$-submodular inequalities, for the hypograph of any $k$-submodular function. This class of inequalities serves as a novel generalization of the well-known submodular inequalities. We show that maximizing a $k$-submodular function is equivalent to solving a mixed-integer linear program with exponentially many $k$-submodular inequalities. Using this representation in a delayed constraint generation framework, we design the first exact algorithm, that is not a complete enumeration method, to solve general $k$-submodular maximization problems. Our computational experiments on the multi-type sensor placement problems demonstrate the efficiency of our algorithm in constrained nonlinear $k$-submodular maximization problems for which no alternative compact mixed-integer linear formulations are available. The computational experiments show that our algorithm significantly outperforms the only available exact solution method—exhaustive search. Problems that would require over 13 years to solve by exhaustive search can be solved within ten minutes using our method.