Improved Parallel Algorithm for Minimum Cost Submodular Cover Problem

Abstract: In the minimum cost submodular cover problem (MinSMC), we are given a monotone nondecreasing submodular function $f\colon 2^V \rightarrow \mathbb{Z}<sup>+$, a linear cost function $c: V\rightarrow \mathbb R<sup>{+}$, 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 $$O(\frac{\log km\log k(\log m+\log\log mk)}{\varepsilon<sup>4})$$ 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 $(0,\frac{1}{5})$.