Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints.

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum size of feasible solution. This immediately gives an $O(1/\varepsilon)$-pass $(2+\varepsilon)$-approximation algorithms for monotone submodular maximization in a matroid and $(3+\varepsilon)$-approximation for monotone submodular matching. Our algorithm is oblivious to the choice $\varepsilon$ and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithm for general, non-negative submodular functions subject to a $p$-matchoid constraint with a number of passes independent of the size of the ground set and $k$. We show that a randomized $O(p/\varepsilon)$-pass algorithm storing $O(p^3k\log(k)/\varepsilon^3)$ elements gives a $(p+1+\bar{\gamma}+O(\varepsilon))$-approximation, where $\bar{gamma}$ is the guarantee of the best-known offline algorithm for the same problem.