Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint

• Published in: Algorithmica Vol. 83; no. 3; pp. 879 - 902
• Main Authors: Huang, Chien-Chung, Kakimura, Naonori
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2021; Springer Verlag
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Algorithms and Data Structures (WADS 2019); Computational Geometry; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Algorithms; Data Structures and Information Theory; Mathematics of Computing; Theory of Computation; Streaming algorithm; Approximation algorithm; Submodular functions
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-020-00786-4

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a ( 0.5 - ε ) -approximate solution in O ( K ε - 1 ) space, where K is the knapsack capacity (Badanidiyuru et al.  KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al.  STOC 2020). In this work, we propose a ( 0.4 - ε ) -approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and ε . This improves on the previous best ratio of 0.363 - ε with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.