A Parallel Algorithm for Minimum Cost Submodular Cover

In a minimum cost submodular cover problem (MinSMC), given a monotone non-decreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}^+$, a cost function $c: V\rightarrow \mathbb R^{+}$, an integer $k\leq f(V)$, the goal is to find a subset $A\subseteq V$ with the minimum cost such that $f(A)\geq k$. MinSMC has a lot of applications in machine learning and data mining. In this paper, we design a parallel algorithm for MinSMC which obtains a solution with approximation ratio at most $\frac{H(\min\{\Delta,k\})}{1-5\varepsilon}$ with probability $1-3\varepsilon$ in $O(\frac{\log m\log n\log^2 mn}{\varepsilon^4})$ rounds, where $\Delta=\max_{v\in V}f(v)$, $H(\cdot)$ is the Hamornic number, $n=f(V)$, $m=|V|$ and $\varepsilon$ is a constant in $(0,\frac{1}{5})$. This is the first paper obtaining a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to $H(\min\{\Delta,k\})$.