Parametric Streaming Two-Stage Submodular Maximization

Abstract

We study the submodular maximization problem in generalized streaming setting using a two-stage policy. In the streaming context, elements are released in a fashion that an element is revealed at one time. Subject to a limited memory capacity, the problem aims to sieve a subset of elements with a sublinear size $\ell$, such that the expecting objective value of all utility functions over the summarized subsets has a performance guarantee. We present a generalized one pass, $\left( \gamma ^5_{\min }/(5+ 2\gamma ^2_{\min } )-O(\epsilon )\right)$-approximation, which consumes $O(\epsilon ^{-1}\ell \log (\ell \gamma _{\min }^{-1}))$ memory and runs in $O(\epsilon ^{-1}kmn\log (\ell \gamma _{\min }^{-1}))$ time, where k, n, m and $\gamma _{\min }$ denote the cardinality constraint, the element stream size, the amount of the learned functions, and the minimum generic submodular ratio of the learned functions, respectively.