Document

# Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case

## File

LIPIcs.ESA.2022.23.pdf
• Filesize: 0.82 MB
• 17 pages

## Cite As

Kobi Bodek and Moran Feldman. Maximizing Sums of Non-Monotone Submodular and Linear Functions: Understanding the Unconstrained Case. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.23

## Abstract

Motivated by practical applications, recent works have considered maximization of sums of a submodular function g and a linear function 𝓁. Almost all such works, to date, studied only the special case of this problem in which g is also guaranteed to be monotone. Therefore, in this paper we systematically study the simplest version of this problem in which g is allowed to be non-monotone, namely the unconstrained variant, which we term Regularized Unconstrained Submodular Maximization (RegularizedUSM). Our main algorithmic result is the first non-trivial guarantee for general RegularizedUSM. For the special case of RegularizedUSM in which the linear function 𝓁 is non-positive, we prove two inapproximability results, showing that the algorithmic result implied for this case by previous works is not far from optimal. Finally, we reanalyze the known Double Greedy algorithm to obtain improved guarantees for the special case of RegularizedUSM in which the linear function 𝓁 is non-negative; and we complement these guarantees by showing that it is not possible to obtain (1/2, 1)-approximation for this case (despite intuitive arguments suggesting that this approximation guarantee is natural).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
• Mathematics of computing → Combinatorial optimization
##### Keywords
• Unconstrained submodular maximization
• regularization
• double greedy
• non-oblivious local search
• inapproximability

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Kobi Bodek and Moran Feldman. Maximizing sums of non-monotone submodular and linear functions: Understanding the unconstrained case. CoRR, abs/2204.03412, 2022. URL: https://doi.org/10.48550/arXiv.2204.03412.
2. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM J. Comput., 44(5):1384-1402, 2015. URL: https://doi.org/10.1137/130929205.
3. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
4. Michele Conforti and Gérard Cornuéjols. Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the rado-edmonds theorem. Discret. Appl. Math., 7(3):251-274, 1984. URL: https://doi.org/10.1016/0166-218X(84)90003-9.
5. Shahar Dobzinski and Jan Vondrák. From query complexity to computational complexity. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference (STOC), pages 1107-1116. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214076.
6. Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM J. Comput., 40(4):1133-1153, 2011. URL: https://doi.org/10.1137/090779346.
7. Moran Feldman. Guess free maximization of submodular and linear sums. Algorithmica, 83(3):853-878, 2021. URL: https://doi.org/10.1007/s00453-020-00757-9.
8. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 570-579. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/FOCS.2011.46.
9. Yuval Filmus and Justin Ward. Monotone submodular maximization over a matroid via non-oblivious local search. SIAM J. Comput., 43(2):514-542, 2014. URL: https://doi.org/10.1137/130920277.
10. Chris Harshaw, Moran Feldman, Justin Ward, and Amin Karbasi. Submodular maximization beyond non-negativity: Guarantees, fast algorithms, and applications. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning (ICML), volume 97 of Proceedings of Machine Learning Research, pages 2634-2643. PMLR, 2019. URL: http://proceedings.mlr.press/v97/harshaw19a.html.
11. Ehsan Kazemi, Shervin Minaee, Moran Feldman, and Amin Karbasi. Regularized submodular maximization at scale. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning (ICML), volume 139 of Proceedings of Machine Learning Research, pages 5356-5366. PMLR, 2021. URL: http://proceedings.mlr.press/v139/kazemi21a.html.
12. Cheng Lu, Wenguo Yang, and Suixiang Gao. Regularized non-monotone submodular maximization. CoRR, abs/2103.10008, 2021. URL: http://arxiv.org/abs/2103.10008.
13. G. L. Nemhauser and L. A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of Operations Research, 3(3):177-188, 1978.
14. Sofia Maria Nikolakaki, Alina Ene, and Evimaria Terzi. An efficient framework for balancing submodularity and cost. In Feida Zhu, Beng Chin Ooi, and Chunyan Miao, editors, The 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), pages 1256-1266. ACM, 2021. URL: https://doi.org/10.1145/3447548.3467367.
15. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Dana Randall, editor, ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1098-1116. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.83.
16. Xin Sun, Dachuan Xu, Yang Zhou, and Chenchen Wu. Maximizing modular plus non-monotone submodular functions. CoRR, abs/2203.07711, 2022. URL: http://arxiv.org/abs/2203.07711.
17. Maxim Sviridenko, Jan Vondrák, and Justin Ward. Optimal approximation for submodular and supermodular optimization with bounded curvature. Math. Oper. Res., 42(4):1197-1218, 2017. URL: https://doi.org/10.1287/moor.2016.0842.
18. Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM J. Comput., 42(1):265-304, 2013. URL: https://doi.org/10.1137/110832318.