Approximation Algorithms for Size-Constrained Non-Monotone Submodular Maximization in Deterministic Linear Time

TL;DR

This paper introduces the first deterministic linear-time algorithms for non-monotone submodular maximization under size constraints, achieving near-optimal approximation ratios in various computational models.

Contribution

It presents new deterministic, linear-time algorithms for non-monotone submodular maximization with size constraints, including streaming and near-linear time solutions.

Findings

First linear-time streaming algorithm with ratio 23.313 + ε

Simpler deterministic linear-time algorithm with ratio 11.657

Deterministic algorithm with ratio e + ε in O_ε(n log n) time

Abstract

In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and nonlinear regression. We provide the first deterministic, linear-time approximation algorithms for this problem that do not assume the objective is monotone. We present three deterministic, linear-time algorithms: a single-pass streaming algorithm with a ratio of $23.313 + \epsilon$, which is the first linear-time streaming algorithm; a simpler deterministic linear-time algorithm with a ratio of $11.657$; and a $(4 + O(\epsilon ))$-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of $e + \epsilon$ in $O_{\epsilon}(n \log(n))$ time, close to the best known expected ratio of $e - 0.121$ in polynomial time.