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

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 + ε, which is the first linear-time streaming algorithm; a simpler deterministic linear-time algorithm with a ratio of 11.657; and a (4 + O(ε))-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of e + ε in O_ε (n log(n)) time, close to the best known expected ratio of e - 0.121 in polynomial time.