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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?

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    $\begingroup$ The result of Feige, Mirrokni and Vondrak applies only to non-negative functions. Your function is no guaranteed to be non-negative unless you are making additional assumptions. I guess you meant the approximation in FMS is 2/5, not 2/4. There is an upcoming paper in FOCS 2012 by Buchbinder etal who obtain an optimal 1/2 approximation for non-negative submodular function maximization. Without non-negativity the problem is inapproximable. See the following paper for some related work on your problem. dl.acm.org/citation.cfm?id=1616497.1616507 $\endgroup$ – Chandra Chekuri Aug 26 '12 at 23:40
  • $\begingroup$ Thanks Chandra for your very helpful comment. You were right about the bound. It has been edited. $\endgroup$ – Ali Aug 27 '12 at 19:52
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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

$$ \sum_{i \in B\setminus A}f(A \cup \{i\}) - f(A) \ge \gamma (f(A \cup B) - f(A)),\;\forall A,B \subseteq [n]. $$

Then, greedy maximization produces a subset $A^G$ with $k$ elements such that

$$ f(A^G)-h(A^G) \ge (1-e^{-\gamma})f(A^*)-h(A^*), $$

where $A^*$ is the $k$-element subset of $[n]$ which maximizes $f(A)-h(A)$. This is a direct consequence of Theorem 3 of this ICML paper.

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