Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one by one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is tomaximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a 1/e-competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most k elements in its solution.
Bibliographical noteFunding Information:
An extended abstract of this work appeared in SODA 2015. The research of Niv Buchbinder was supported by ISF grant 1585/15 and USIsrael BSF grant 2014414. The research of Moran Feldman was supported in part by ERC Starting Grant 335288-OptApprox and ISF grant 1357/16. The research of Roy Schwartz was supported in part by ISF grant 1336/16. Authors’ addresses: N. Buchbinder, Tel Aviv University, Statistics and Operations Research Department, Tel Aviv, Israel; email: firstname.lastname@example.org; M. Feldman, The Open University of Israel, Department of Mathematics and Computer Science, Raanana, Israel; email: email@example.com; R. Schwartz, Technion, Computer Science Department, Haifa, Israel; email: firstname.lastname@example.org. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from email@example.com. © 2019 Association for Computing Machinery. 1549-6325/2019/05-ART30 $15.00 https://doi.org/10.1145/3309764
© 2019 Association for Computing Machinery.
- Competitive analysis
- Online algorithms
- Submodular maximization