Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization

We consider maximizing a monotone submodular function under a cardinality constraint or a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access to only a small fraction of the data stored in primary memory. We propose the following streaming algorithms taking O(ε^{− 1}) passes: (1) a (1 − e^{− 1} − ε)-approximation algorithm for the cardinality-constrained problem, (2) a (0.5 − ε)-approximation algorithm for the knapsack-constrained problem. Both of our algorithms run deterministically in O^∗(n) time, using O^∗(K) space, where n is the size of the ground set and K is the size of the knapsack. Here the term O^∗ hides a polynomial of logK and ε^{− 1}. Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes O(nε^{− 1}log(ε^{− 1}logK)) time, improving on the algorithm of Badanidiyuru and Vondrák that takes O(nε^{− 1}log(ε^{− 1}K)) time.