Randomization is a fundamental tool used in many theoretical and practical areas of computer science. We study here the role of randomization in the area of submodular function maximization. In this area, most algorithms are randomized, and in almost all cases the approximation ratios obtained by current randomized algorithms are superior to the best results obtained by known deterministic algorithms. Derandomization of algorithms for general submodular function maximization seems hard since the access to the function is done via a value oracle. This makes it hard, for example, to apply standard derandomization techniques such as conditional expectations. Therefore, an interesting fundamental problem in this area is whether randomization is inherently necessary for obtaining good approximation ratios. In this work, we give evidence that randomization is not necessary for obtaining good algorithms by presenting a new technique for derandomization of algorithms for submodular function maximization. Our high level idea is to maintain explicitly a (small) distribution over the states of the algorithm, and carefully update it using marginal values obtained from an extreme point solution of a suitable linear formulation. We demonstrate our technique on two recent algorithms for unconstrained submodular maximization and for maximizing a submodular function subject to a cardinality constraint. In particular, for unconstrained submodular maximization we obtain an optimal deterministic 1/2-approximation showing that randomization is unnecessary for obtaining optimal results for this setting.
|Journal||ACM Transactions on Algorithms|
|State||Published - Jul 2018|
Bibliographical noteFunding Information:
The work of Niv Buchbinder is supported by ISF grant number 1585/15 and BSF grant number 2014414. The work of Moran Feldman was supported in part by ERC Starting Grant 335288-OptApprox and ISF grant number 1357/16. A preliminary version of this paper appeared in the proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA16). Authors’ addresses: N. Buchbinder, Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv, Israel 39040; email: firstname.lastname@example.org; M. Feldman, Department of Mathematics and Computer Science, The Open University of Israel, Raanana 4353701, Israel; email: email@example.com. 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 firstname.lastname@example.org. © 2018 ACM 1549-6325/2018/06-ART32 $15.00 https://doi.org/10.1145/3184990
- Submodular function maximization
- Unconstrained submodular maximization