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(epsilon(-1)) passes: (1) a (1-e(-1)-epsilon)-approximation algorithm for the cardinality-constrained problem, (2) a (0.5 - epsilon)-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 log K and epsilon(-1). Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes O (n epsilon(-1) log (epsilon(-1) log K)) time, improving on the algorithm of Badanidiyuru and Vondrak that takes O (n epsilon(-1)log (epsilon(-1) K)) time.