Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization
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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: ----a (1-e^-1-ε)-approximation algorithm for the cardinality-constrained problem ---- 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 and K is the size of the knapsack. Here the term O^∗ hides a polynomial of 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^-1 (ε^-1 K) ) time, improving on the algorithm of Badanidiyuru and Vondrák that takes O(n ε^-1 (ε^-1 K) ) time.
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