A Semi-streaming Algorithm for Monotone Regularized Submodular Maximization with a Matroid Constraint

Abstract

In the face of large-scale datasets in many practical problems, it is an effective method to design approximation algorithms for maximizing a regularized submodular function in a semi-streaming model. In this paper, we study the monotone regularized submodular maximization with a matroid constraint and present a single-pass semi-streaming algorithm using multilinear extension function and greedy idea. We show that our algorithm has an approximation ratio of $(\frac{(\beta -1)(1-\textrm{e}^{-\alpha })}{\beta +\alpha \beta -\alpha }, \frac{\beta (\beta -1)(1-\textrm{e}^{-\alpha })}{\beta +\alpha \beta -\alpha })$ and a memory of O(r(M)), where r(M) is the rank of the matroid and parameters $\alpha ,\beta >1$. Specifically, if $\alpha =1.18$ and $\beta = 9.784$, our algorithm is (0.302, 2.955)-approximate. If $\alpha =1.257$ and $\beta = 3.669$, our algorithm is $(0.272\, 6,1)$-approximate.