Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization

• Published in: Theory of computing systems Vol. 66; no. 1; pp. 354 - 394
• Main Authors: Huang, Chien-Chung, Kakimura, Naonori
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2022; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Approximation; Computer Science; Constraints; Mathematical analysis; Maximization; Optimization; Polynomials; Theory of Computation; Submodular function maximization; Streaming algorithms; Approximation algorithms
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-021-10065-6

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