I'm considering a network planning problem which is stated as follows: From the given ground set $\mathcal{V}$, select $\mathcal{A} \subseteq \mathcal{V}$ such that \begin{equation} f(\mathcal{A}) - \sum_{v_i \in \mathcal{A}} c_i \end{equation} is maximized where $f$ is a monotone submodular function and $c_i \ge 0$ is the cost of selecting $v_i$. The problem is an instance of nonomonotone submodular maximization for which a local search heuristic with approximation bound of $\frac{2}{5} - \frac{\epsilon}{n}$ is presented in
Uriel Feige, Vahab S. Mirrokni, Jan Vondrák: "Maximizing Non-Monotone Submodular Functions ", FOCS 2007.
I'm wondering if anyone is aware of a better approximation algorithm for my specific problem?