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 log K 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 ^-1log (ε ^-1log K) ) time, improving on the algorithm of Badanidiyuru and Vondrák that takes O(n ε ^-1log (ε ^-1 K) ) time.