Instance Specific Approximations for Submodular Maximization

Eric Balkanski, Sharon Qian, Yaron Singer
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:609-618, 2021.

Abstract

The predominant measure for the performance of an algorithm is its worst-case approximation guarantee. While worst-case approximations give desirable robustness guarantees, they can differ significantly from the performance of an algorithm in practice. For the problem of monotone submodular maximization under a cardinality constraint, the greedy algorithm is known to obtain a 1-1/e approximation guarantee, which is optimal for a polynomial-time algorithm. However, very little is known about the approximation achieved by greedy and other submodular maximization algorithms on real instances. We develop an algorithm that gives an instance-specific approximation for any solution of an instance of monotone submodular maximization under a cardinality constraint. This algorithm uses a novel dual approach to submodular maximization. In particular, it relies on the construction of a lower bound to the dual objective that can also be exactly minimized. We use this algorithm to show that on a wide variety of real-world datasets and objectives, greedy and other algorithms find solutions that approximate the optimal solution significantly better than the 1-1/e   0.63 worst-case approximation guarantee, often exceeding 0.9.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-balkanski21a, title = {Instance Specific Approximations for Submodular Maximization}, author = {Balkanski, Eric and Qian, Sharon and Singer, Yaron}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {609--618}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/balkanski21a/balkanski21a.pdf}, url = {https://proceedings.mlr.press/v139/balkanski21a.html}, abstract = {The predominant measure for the performance of an algorithm is its worst-case approximation guarantee. While worst-case approximations give desirable robustness guarantees, they can differ significantly from the performance of an algorithm in practice. For the problem of monotone submodular maximization under a cardinality constraint, the greedy algorithm is known to obtain a 1-1/e approximation guarantee, which is optimal for a polynomial-time algorithm. However, very little is known about the approximation achieved by greedy and other submodular maximization algorithms on real instances. We develop an algorithm that gives an instance-specific approximation for any solution of an instance of monotone submodular maximization under a cardinality constraint. This algorithm uses a novel dual approach to submodular maximization. In particular, it relies on the construction of a lower bound to the dual objective that can also be exactly minimized. We use this algorithm to show that on a wide variety of real-world datasets and objectives, greedy and other algorithms find solutions that approximate the optimal solution significantly better than the 1-1/e   0.63 worst-case approximation guarantee, often exceeding 0.9.} }
Endnote
%0 Conference Paper %T Instance Specific Approximations for Submodular Maximization %A Eric Balkanski %A Sharon Qian %A Yaron Singer %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-balkanski21a %I PMLR %P 609--618 %U https://proceedings.mlr.press/v139/balkanski21a.html %V 139 %X The predominant measure for the performance of an algorithm is its worst-case approximation guarantee. While worst-case approximations give desirable robustness guarantees, they can differ significantly from the performance of an algorithm in practice. For the problem of monotone submodular maximization under a cardinality constraint, the greedy algorithm is known to obtain a 1-1/e approximation guarantee, which is optimal for a polynomial-time algorithm. However, very little is known about the approximation achieved by greedy and other submodular maximization algorithms on real instances. We develop an algorithm that gives an instance-specific approximation for any solution of an instance of monotone submodular maximization under a cardinality constraint. This algorithm uses a novel dual approach to submodular maximization. In particular, it relies on the construction of a lower bound to the dual objective that can also be exactly minimized. We use this algorithm to show that on a wide variety of real-world datasets and objectives, greedy and other algorithms find solutions that approximate the optimal solution significantly better than the 1-1/e   0.63 worst-case approximation guarantee, often exceeding 0.9.
APA
Balkanski, E., Qian, S. & Singer, Y.. (2021). Instance Specific Approximations for Submodular Maximization. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:609-618 Available from https://proceedings.mlr.press/v139/balkanski21a.html.

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