A Multi-pass Streaming Algorithm for Regularized Submodular Maximization

Abstract

In this paper, we consider a problem of maximizing regularized submodular functions with a k-cardinality constraint under streaming fashion. In the model, the utility function $f(\cdot )=g(\cdot )-\ell (\cdot )$ is expressed as the difference between a non-negative monotone non-decreasing submodular function g and a non-negative modular function $\ell$. In addition, the elements are revealed in a streaming setting, that is to say, an element is visited in one time slot. The problem asks to find a subset of size at most k such that the regularized utility value is maximized. Most of the existing algorithms for the submodular maximization heavily rely on the non-negativity assumption of the utility function, which may not be applicable for our regularized scenario. Indeed, determining if the maximum is positive or not is NP-hard, which implies that no multiplicative factor approximation is existed for this problem. To circumvent this challenge, several works paid attention to more meaningful guarantees by introducing a slightly weaker notion of approximation, and any developed algorithm is aim to construct a solution S satisfying $f(S)\ge \rho \cdot g(OPT)-\ell (OPT)$ for some $\rho >0$. In this work, assume there is a weak $\rho$-approximation for the k-cardinality constrained regularized submodular maximization, we develop Distorted-Threshold-Streaming, a multi-pass bicriteria algorithm for the streaming regularized submodular maximization with the k-cardinality constraint (SRSMCC), which produces a $(\rho /\lambda ,1/\lambda )$-bicriteria approximation by making over $O(\log (\lambda /\rho )/\varepsilon )$ passes, consuming O(k) memory and using $O(\log (\lambda /\rho )/\varepsilon )$ queries per element, where $\lambda =\frac{2-(2\rho +2)/(3+\sqrt{5-4\rho })}{(3+\sqrt{5-4\rho })/(2\rho +2)-1}$ and any accuracy parameter $\varepsilon >0$.