In the minimum cost submodular cover problem (MinSMC), we are given a monotone nondecreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}^+$, a linear 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 the MinSMC that takes at most $O(\frac{\log km\log k(\log m+\log\log mk)}{\varepsilon^4})$ adaptive rounds, and it achieves an approximation ratio of $\frac{H(\min\{\Delta,k\})}{1-5\varepsilon}$ with probability at least $1-3\varepsilon$, where $\Delta=\max_{v\in V}f(v)$, $H(\cdot)$ is the Harmonic number, $m=|V|$, and $\varepsilon$ is a constant in $(0,\frac{1}{5})$.
翻译:在最低成本子模块覆盖问题( MinsMC ) 中, 我们得到一个单调的不声明子模块功能 $f\ colom 2 ⁇ V\rightrow \ mathbb $, 线性成本函数 $c: V\rightrow\mathbbR $ $, 整数 k\leq f( V) 美元, 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的核心 。 在本文中, 我们为 MinSMC 设计一个平行的算法, 该算法最多以O(\\ frac\log km\ log k k (log\ mk)\ nk = varvaressléx$.