We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SAMPLEGREEDY, which obtains a (p + 2 + o(1))-approximation for maximizing a submodular function subject to a p-extendible system using O(n + nk=p) evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present SAMPLE-STREAMING, which obtains a (4p + 2 − o(1))-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and O(km=p) evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.
Bibliographical noteFunding Information:
Funding: This work was supported in part by a National Science Foundation (NSF) Graduate Research Fellowship [Grant DGE1122492] awarded to C. Harshaw, an Israel Science Foundation (ISF) [Grant 1357/16] awarded to M. Feldman, and the NSF [Grant IIS-1845032], the Office of Naval Research [Grant N00014-19-2406], and funding from TATA Sons Private Limited awarded to A. Karbasi.
© 2021 INFORMS
- approximation algorithms
- p-extendible systems
- streaming algorithms
- submodular maximization
ASJC Scopus subject areas
- Mathematics (all)
- Computer Science Applications
- Management Science and Operations Research