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) passes: 1. a (1 − e − 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 in O∗(n) time, using O∗(K) space, where n is the size of the ground set andK is the size of the knapsack. Here the term O∗ hides a polynomial of logK and ε. Our streaming algorithms can also be used as fast approximation algorithms. In particular, for the cardinality-constrained problem, our algorithm takes O(nε log(ε logK)) time, improving on the algorithm of Badanidiyuru and Vondrák that takes O(nε log(εK)) time.