In the minimum cost submodular cover problem (MinSMC) problem, given a monotone nondecreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}^+$, a cost function $c: V\rightarrow \mathbb R^{+}$, and an integer $k\leq f(V)$, the goal is to find a subset $A\subseteq V$ with the minimum cost such that $f(A)\geq k$. The MinSMC can be found at the heart of many machine learning and data mining applications. In this paper, we design a parallel algorithm for MinSMC that obtains a solution with an approximation ratio of at most $\frac{H(\min\{\Delta,k\})}{1-5\varepsilon}$ with a probability of $1-3\varepsilon$ in $O(\frac{\log km\log k(\log m+\log\log mk)}{\varepsilon^4})$ adaptive rounds, where $\Delta=\max_{v\in V}f(v)$, $H(\cdot)$ is the Harmonic number, $m=|V|$, and $\varepsilon$ is a constant in $(0,\frac{1}{5})$. This paper is the first to obtain a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to $H(\min\{\Delta,k\})$.
翻译：在最低成本子模块覆盖问题( MinSMC ) 问题中, 考虑到单调的未分解子模块功能 $f\ cron 2 ⁇ V\rightrow\mathb}$, 成本函数 $c: V\rightrow\mathbbR $, 和整数 $k\leq f( V), 目标是找到一个子 $A\ subseq V$, 其最低成本为 $f( A)\ geq k$。 MinSMC 位于许多机器学习和数据挖掘应用程序的核心 。 在本文中, 我们为 MinSMC 设计一个平行的平行算法, 它获得的解决方案, 接近率最高为$\\\\\\\\\ m\\\\\\\\\\\\\\\\\\\\\\ f( m), 1-3\ varepsilon$ (\\ log) k k) k。 (\\\ log m) m\\\\\ fm) lexn courn road回合, $ Delate $@xxx__xxxxxx_____xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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