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To simplify life, let $\mathcal V = [n] := \{1,2,\ldots,n\}$. For $A \subseteq [n]$, define $h(A):=\sum_{i \in A}c_i$. Note that $h$ defines a modular (i.e additive) set function. Now, suppose there exists $\gamma \in [0, 1]$ such that $f$ is weakly $\gamma$-submodular, i.e such that $$ \sum_{i \in B\setminus A}f(A \cup \{i\}) - f(A) \ge \gamma (f(A \cup B)...


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