On Maximizing Sums of Non-Monotone Submodular and Linear Functions

Author Benjamin Qi

Thumbnail PDF


  • Filesize: 0.77 MB
  • 16 pages

Document Identifiers

Author Details

Benjamin Qi
  • Massachusetts Institute of Technology, Cambridge, MA, USA


I thank Tasuku Soma for mentoring me through the MIT Undergraduate Research Opportunities Program, as well as Moran Feldman for providing many helpful suggestions.

Cite AsGet BibTex

Benjamin Qi. On Maximizing Sums of Non-Monotone Submodular and Linear Functions. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by [Bodek and Feldman '22]. In this problem, we are given query access to a non-negative submodular function f: 2^N → ℝ_{≥ 0} and a linear function 𝓁: 2^N → ℝ over the same ground set N, and the objective is to output a set T ⊆ N approximately maximizing the sum f(T)+𝓁(T). Specifically, an algorithm is said to provide an (α,β)-approximation for RegularizedUSM if it outputs a set T such that E[f(T)+𝓁(T)] ≥ max_{S ⊆ N}[α ⋅ f(S)+β⋅ 𝓁(S)]. We also study the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied [Sviridenko et al. '17, Feldman '18, Harshaw et al. '19]. On the other hand, we are aware of only one prior work that studies RegularizedCSM with non-monotone f [Lu et al. '21], and that work constrains 𝓁 to be non-positive. In this work, we provide improved (α,β)-approximation algorithms for both {RegularizedUSM} and {RegularizedCSM} with non-monotone f. In particular, we are the first to provide nontrivial (α,β)-approximations for RegularizedCSM where the sign of 𝓁 is unconstrained, and the α we obtain for RegularizedUSM improves over [Bodek and Feldman '22] for all β ∈ (0,1). In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the α our algorithm obtains for {RegularizedCSM} with unconstrained 𝓁 is essentially tight for β ≥ e/(e+1). Using similar ideas, we are also able to show 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to [Oveis Gharan and Vondrak '10].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Approximation algorithms analysis
  • submodular maximization
  • regularization
  • continuous greedy
  • inapproximability


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


  1. Kobi Bodek and Moran Feldman. Maximizing sums of non-monotone submodular and linear functions: Understanding the unconstrained case, 2022. To appear in ESA 2022. URL: https://doi.org/10.48550/ARXIV.2204.03412.
  2. Niv Buchbinder and Moran Feldman. Constrained submodular maximization via a nonsymmetric technique. Mathematics of Operations Research, 44(3):988-1005, 2019. Google Scholar
  3. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 649-658, 2012. URL: https://doi.org/10.1109/FOCS.2012.73.
  4. Gruia Calinescu, Chandra Chekuri, Martin Pal, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. Google Scholar
  5. Uriel Feige, Vahab S Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM Journal on Computing, 40(4):1133-1153, 2011. Google Scholar
  6. Moran Feldman. Guess free maximization of submodular and linear sums. Algorithmica, 83(3):853-878, 2021. Google Scholar
  7. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 570-579. IEEE, 2011. Google Scholar
  8. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1098-1116. SIAM, 2011. Google Scholar
  9. Michael Gygli, Helmut Grabner, and Luc Van Gool. Video summarization by learning submodular mixtures of objectives. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3090-3098, 2015. Google Scholar
  10. Chris Harshaw, Moran Feldman, Justin Ward, and Amin Karbasi. Submodular maximization beyond non-negativity: Guarantees, fast algorithms, and applications. In International Conference on Machine Learning, pages 2634-2643. PMLR, 2019. Google Scholar
  11. Stefanie Jegelka and Jeff Bilmes. Submodularity beyond submodular energies: coupling edges in graph cuts. In CVPR 2011, pages 1897-1904. IEEE, 2011. Google Scholar
  12. Ehsan Kazemi, Shervin Minaee, Moran Feldman, and Amin Karbasi. Regularized submodular maximization at scale. In International Conference on Machine Learning, pages 5356-5366. PMLR, 2021. Google Scholar
  13. David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 137-146, 2003. Google Scholar
  14. Andreas Krause and Daniel Golovin. Submodular function maximization. Tractability, 3:71-104, 2014. Google Scholar
  15. Andreas Krause and Carlos Guestrin. Submodularity and its applications in optimized information gathering. ACM Transactions on Intelligent Systems and Technology (TIST), 2(4):1-20, 2011. Google Scholar
  16. Andreas Krause, Ajit Singh, and Carlos Guestrin. Near-optimal sensor placements in gaussian processes: Theory, efficient algorithms and empirical studies. Journal of Machine Learning Research, 9(2), 2008. Google Scholar
  17. Hui Lin and Jeff Bilmes. A class of submodular functions for document summarization. In Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, pages 510-520, 2011. Google Scholar
  18. Cheng Lu, Wenguo Yang, and Suixiang Gao. Regularized non-monotone submodular maximization, 2021. URL: https://doi.org/10.48550/ARXIV.2103.10008.
  19. George L Nemhauser and Laurence A Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of operations research, 3(3):177-188, 1978. Google Scholar
  20. George L Nemhauser, Laurence A Wolsey, and Marshall L Fisher. An analysis of approximations for maximizing submodular set functions—i. Mathematical programming, 14(1):265-294, 1978. Google Scholar
  21. Sofia Maria Nikolakaki, Alina Ene, and Evimaria Terzi. An efficient framework for balancing submodularity and cost. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 1256-1266, 2021. Google Scholar
  22. Benjamin Qi. On maximizing sums of non-monotone submodular and linear functions, 2022. URL: https://doi.org/10.48550/ARXIV.2205.15874.
  23. Jianbing Shen, Zhiyuan Liang, Jianhong Liu, Hanqiu Sun, Ling Shao, and Dacheng Tao. Multiobject tracking by submodular optimization. IEEE transactions on cybernetics, 49(6):1990-2001, 2018. Google Scholar
  24. Maxim Sviridenko, Jan Vondrák, and Justin Ward. Optimal approximation for submodular and supermodular optimization with bounded curvature. Mathematics of Operations Research, 42(4):1197-1218, 2017. Google Scholar
  25. Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM Journal on Computing, 42(1):265-304, 2013. Google Scholar
  26. Kai Wei, Yuzong Liu, Katrin Kirchhoff, and Jeff Bilmes. Using document summarization techniques for speech data subset selection. In Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 721-726, 2013. Google Scholar