Improved Parallel Algorithm for Minimum Cost Submodular Cover Problem

TL;DR

This paper presents a new parallel algorithm for the minimum cost submodular cover problem, achieving faster adaptive rounds and near-optimal approximation ratios, which is significant for applications in machine learning and data mining.

Contribution

The paper introduces a novel parallel algorithm for MinSMC with improved adaptive complexity and approximation guarantees compared to previous methods.

Findings

Achieves $O(rac{ ext{log} km ext{log} k( ext{log} m+ ext{log} ext{log} mk)}{ ext{varepsilon}^4})$ adaptive rounds.

Provides an approximation ratio of $rac{H( ext{min}\{ ext{ extDelta},k ext})}{1-5 ext{ extvarepsilon}}$ with high probability.

Applicable to large-scale machine learning and data mining tasks involving submodular functions.

Abstract

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 $(0,\frac{1}{5})$.