Quick Streaming Algorithms for Maximization of Monotone Submodular Functions in Linear Time

We consider the problem of monotone, submodular maximization over a ground set of size $n$ subject to cardinality constraint $k$. For this problem, we introduce streaming algorithms with linearquery complexity and linear number of arithmetic operations; these algorithms are the first deterministic algorithms for submodular maximization that require a linear number of arithmetic operations. Specifically, for any $c \ge 1, \epsilon > 0$, we propose a single-pass, deterministic streaming algorithm with ratio $1/(4c)-\epsilon$, query complexity $\lceil n / c \rceil + c$, memory complexity $O(k \log k)$, and $O(n)$ total running time. As $k \to \infty$, the ratio converges to $(1 - 1/e)/(c + 1)$. In addition, we propose a deterministic, multi-pass streaming algorithm with $O(1 / \epsilon)$ passes that achieves ratio $1-1/e - \epsilon$ in $O(n/\epsilon)$ queries, $O(k \log (k))$ memory, and $O(n)$ time. We prove a lower bound that implies no constant-factor approximation exists using $o(n)$ queries, even if queries to infeasible sets are allowed. An experimental analysis demonstrates that our algorithms require fewer queries (often substantially less than $n$) to achieve better objective value than the current state-of-the-art algorithms.