A note on maximizing the difference between a monotone submodular function and a linear function.

Motivated by team formation applications, we study discrete optimization problems of the form $\max_{S\in\mathcal{S}}\left(f(S)-w(S)\right)$, where $f:2^{V}\to\mathbb{R_{+}}$ is a non-negative monotone submodular function, $w:2^{V}\to\mathbb{R}_{+}$ is a non-negative linear function, and $\mathcal{S}\subseteq2^{V}$. We give very simple and efficient algorithms for classical constraints, such as cardinality and matroid, that work in a variety of models, including the offline, online, and streaming. Our algorithms use a very simple scaling approach: we pick an absolute constant $c\geq1$ and optimize the function $f(S)-c\cdot w(S)$ using a black-box application of standard algorithms, such as the classical Greedy algorithm and the single-threshold Greedy algorithm. These algorithms are based on recent works that use (time varying) scaling combined with classical algorithms such as the discrete and continuous Greedy algorithms (Feldman, WADS'19; Harshaw \emph{et al.}, ICML'19).