A Reduction for Optimizing Lattice Submodular Functions with Diminishing Returns

A function $f: \mathbb{Z}_+^E \rightarrow \mathbb{R}_+$ is DR-submodular if it satisfies $f(\bx + \chi_i) -f (\bx) \ge f(\by + \chi_i) - f(\by)$ for all $\bx\le \by, i\in E$. Recently, the problem of maximizing a DR-submodular function $f: \mathbb{Z}_+^E \rightarrow \mathbb{R}_+$ subject to a budget constraint $\|\bx\|_1 \leq B$ as well as additional constraints has received significant attention \cite{SKIK14,SY15,MYK15,SY16}. In this note, we give a generic reduction from the DR-submodular setting to the submodular setting. The running time of the reduction and the size of the resulting submodular instance depends only \emph{logarithmically} on $B$. Using this reduction, one can translate the results for unconstrained and constrained submodular maximization to the DR-submodular setting for many types of constraints in a unified manner.