Simultaenous Sieves: A Deterministic Streaming Algorithm for Non-Monotone Submodular Maximization

TL;DR

This paper introduces a deterministic streaming algorithm for non-monotone submodular maximization that nearly matches the best randomized algorithms, improving approximation ratios significantly in polynomial time.

Contribution

It presents a novel deterministic single-pass streaming algorithm for non-monotone submodular maximization with improved approximation guarantees.

Findings

Achieves approximation ratio close to 0.2689 in polynomial time.

Reduces to optimal algorithm for monotone case.

Nearly matches randomized algorithms' ratios in expectation.

Abstract

In this work, we present a combinatorial, deterministic single-pass streaming algorithm for the problem of maximizing a submodular function, not necessarily monotone, with respect to a cardinality constraint (SMCC). In the case the function is monotone, our algorithm reduces to the optimal streaming algorithm of Badanidiyuru et al. (2014). In general, our algorithm achieves ratio $\alpha / (1 + \alpha) - \varepsilon$, for any $\varepsilon > 0$, where $\alpha$ is the ratio of an offline (deterministic) algorithm for SMCC used for post-processing. Thus, if exponential computation time is allowed, our algorithm deterministically achieves nearly the optimal $1/2$ ratio. These results nearly match those of a recently proposed, randomized streaming algorithm that achieves the same ratios in expectation. For a deterministic, single-pass streaming algorithm, our algorithm achieves in polynomial time an improvement of the best approximation factor from $1/9$ of previous literature to $\approx 0.2689$.