Skip to main content
Springer Nature Link
Log in

Sequence submodular maximization meets streaming

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

In this paper, we study the problem of maximizing a sequence submodular function in the streaming setting, where the utility function is defined on sequences instead of sets of elements. We encode the sequence submodular maximization with a weighted digraph, in which the weight of a vertex reveals the utility value in selecting a single element and the weight of an edge reveals the additional profit with respect to a certain selection sequence. The edges are visited in a streaming fashion and the aim is to sieve a sequence of at most k elements from the stream, such that the utility is maximized. In this work, we present an edge-based threshold procedure, which makes one pass over the stream, attains an approximation ratio of \((1/(2\varDelta +1)- O(\epsilon ))\), consumes \(O(k\varDelta /\epsilon )\) memory source in total and \(O(\log (k\varDelta )/\epsilon )\) update time per edge, where \(\varDelta \) is the minimum of the maximal outdegree and indegree of the directed graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Alaluf N, Feldman M (2019) Making a sieve random: Improved semi-streaming algorithm for submodular maximization under a cardinality constraint. ArXiv: 1906.11237

  • Badanidiyuru A, Mirzasoleiman B, Karbasi A, Krause A (2014) Streaming submodular maximization: Massive data summarization on the fly. In Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining, pp 671–680

  • Bai W, Bilmes J (2018) Greed is still good: Maximizing monotone submodular \(+\) supermodular(BP) functions. In Proceedings of the 35th international conference on machine learning, pp 314–323

  • Bian AA, Buhmann JM, Krause A, Tschiatschek S (2017) Guarantees for greedy maximization of non-submodular functions with applications. In Proceedings of the 34th international conference on machine learning, pp 498–507

  • Bogunovic I, Zhao J, Cevher V (2018) Robust maximization of non-submodular objectives. In Proceedings of the 21st international conference on artificial intelligence and statistics, pp 890–899

  • Buchbinder N, Feldman M, Garg M (2019) Deterministic \((1/2+\epsilon )\)-approximation for submodular maximization over a matroid. In Proceedings of the 30th Annual ACM-SIAM symposium on discrete algorithms, pp 241–254

  • Buchbinder N, Feldman M, Naor JS, Schwartz R (2014) Submodular maximization with cardinality constraints. In Proceedings of the 25th Annual ACM-SIAM symposium on discrete algorithms, pp 1433–1452

  • Buchbinder N, Feldman M, Seffi J, Schwartz R (2015) A tight linear time \((1/2)\)-approximation for unconstrained submodular maximization. SIAM J Comput 44(5):1384–1402

    Article  MathSciNet  Google Scholar 

  • Calinescu G, Chekuri C, Pal M, Vondrák J (2011) Maximizing a monotone submodular function subject to a matroid constraint. SIAM J Comput 40(6):1740–1766

    Article  MathSciNet  Google Scholar 

  • Chakrabarti A, Kale S (2015) Submodular maximization meets streaming: Matchings, matroids, and more. Math Program 154(1–2):225–247

    Article  MathSciNet  Google Scholar 

  • Chekuri C, Gupta S, Quanrud K (2015) Streaming algorithms for submodular function maximization. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming, pp 318–330

  • Conforti M, Cornuéjols G (1984) Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discrete Appl Math 7(3):251–274

    Article  MathSciNet  Google Scholar 

  • Das A, Kempe D (2011) Submodular meets spectral: Greedy algorithms for subset selection, sparse approximation and dictionary selection. In Proceedings of the 28th international conference on machine learning, pp 1057–1064

  • Du D, Li Y, Xiu N, Xu D (2014) Simultaneous approximation of multi-criteria submodular function maximization. J Oper Res Soc China 2(3):271–290

    Article  MathSciNet  Google Scholar 

  • Elenberg ER, Dimakis AG, Feldman M, Karbasi A (2017) Streaming weak submodularity: Interpreting neural networks on the fly. In Proceedings of the 31st international conference on neural information processing systems, pp 4044–4054

  • Ene A, Nguyen HL (2016) Constrained submodular maximization: Beyond \(1/e\). In Proceedings of the 57th IEEE annual symposium on foundations of computer science, pp 248–257

  • Ene A, Nguyen HL, Suh A (2019) An optimal streaming algorithm for non-monotone submodular Maximization. ArXiv:1911.12959

  • Feige U (1998) A threshold of \(\ln n\) for approximating set cover. J ACM 45(4):634–652

    Article  MathSciNet  Google Scholar 

  • Feige U, Izsak R (2013) Welfare maximization and the supermodular degree. In Proceedings of the 4th conference on innovations in theoretical computer science, pp 247-256

  • Feige U, Mirrokni VS, Vondrák J (2011) Maximizing non-monotone submodular functions. SIAM J Comput 40(4):1133–1153

    Article  MathSciNet  Google Scholar 

  • Feldman M, Karbasi A, Kazemi E (2018) Do less, get more: Streaming submodular maximization with subsampling. In Proceedings of the 32nd international conference on neural information processing systems, pp 730–740

  • Feldman M, Naor J, Schwartz R (2011) A unified continuous greedy algorithm for submodular maximization. In Proceedings of the 52nd IEEE annual symposium on foundations of computer science, pp 570–579

  • Feldman M, Naor JS, Schwartz R (2011) Nonmonotone submodular maximization via a structural continuous greedy algorithm. In Proceedings of the 38th international colloquium on automata, languages, and programming, pp 342–353

  • Filmus Y, Ward J (2012) A tight combinatorial algorithm for submodular maximization subject to a matroid constraint. In Proceedings of the 53rd Annual IEEE symposium on foundations of computer science, pp 659–668

  • Gharan SO, Vondrák J (2011) Submodular maximization by simulated annealing. In Proceedings of the 22nd Annual ACM-SIAM symposium on discrete algorithms, pp 1098–1116

  • Golovin D, Krause A (2011) Adaptive submodularity: theory and applications in active learning and stochastic optimization. J Artif Intell Res 42(1):427–486

    MathSciNet  Google Scholar 

  • Gupta A, Roth A, Schoenebeck G, Talwar K (2010) Constrained non-monotone submodular maximization: offline and secretary algorithms. In Proceedings of the 6th international conference on internet and network economics, pp 246–257

  • Haba R, Kazemi E, Feldman F, Karbasi A (2020) Streaming submodular maximization under a \(k\)-set system constraint. ArXiv: 2002.03352

  • Huang CC, Kakimura N, Yoshida Y (2017) Streaming algorithms for maximizing monotone submodular functions under a knapsack constraint. In Proceedings of the 20th international workshop on approximation algorithms for combinatorial optimization problems and the 21st international workshop on randomization and computation No. 11, 11:1–11:14

  • Jiang Y, Wang Y, Xu D, Yang R, Zhang Y (2019) Streaming algorithm for maximizing a monotone non-submodular function under \(d\)-knapsack constraint. Optim Lett. https://doi.org/10.1007/s11590-019-01430-z

    Article  Google Scholar 

  • Kazemi E, Mitrovic M, Zadimoghaddam M, Lattanzi S, Karbasi A (2019) Submodular streaming in all its glory: Tight approximation, minimum memory and low adaptive complexity. In Proceedings of the 36th international conference on machine learning, pp 3311–3320

  • Lee J, Mirrokni VS, Nagarajan V, Sviridenko M (2010) Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J Discrete Math 23(4):2053–2078

    Article  MathSciNet  Google Scholar 

  • Mitrovic M, Feldman M, Krause A, Karbasi A (2018) Submodularity on hypergraphs: From sets to sequences. In Proceedings of the 21st international conference on artificial intelligence and statistics, pp 1177–1184

  • Mitrovic M, Kazemi E, Feldman M, Krause A, Karbasi A (2019) Adaptive sequence submodularity. ArXiv:1902.05981

  • Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions-I. Math Program 14(1):265–294

    Article  MathSciNet  Google Scholar 

  • Norouzi-Fard A, Tarnawski J, Mitrović S, Zandieh A, Mousavifar A, Svensson O (2018) Beyond \(1/2\)-approximation for submodular maximization on massive data streams. In Proceedings of the 35th international conference on machine learning, pp 3826–3835

  • Sviridenko M (2004) A note on maximizing a submodular set function subject to a knapsack constraint. Oper Res Lett 32(1):41–43

    Article  MathSciNet  Google Scholar 

  • Sviridenko M, Vondrák J, Ward J (2015) Optimal approximation for submodular and supermodular optimization with bounded curvature. In: Proceedings of the 26th Annual ACM-SIAM symposium on discrete algorithms, pp 1134–1148

  • Tschiatschek S, Singla A, Krause A (2017) Selecting sequences of items via submodular maximization. In: Proceedings of the 31st AAAI conference on artificial intelligence, pp 2667–2673

  • Wang Z, Moran B, Wang X, Pan Q (2016) Approximation for maximizing monotone non-decreasing set functions with a greedy method. J Comb Optim 31(1):29–43

    Article  MathSciNet  Google Scholar 

  • Wang Y, Xu D, Jiang Y, Zhang D (2019) Minimizing ratio of monotone non-submodular functions. J Oper Res Soc China 7(3):449–459

    Article  MathSciNet  Google Scholar 

  • Wang Y, Xu D, Wang Y, Zhang D (2020) Non-submodular maximization on massive data streams. J Glob Optim 76(4):729–743

    Article  MathSciNet  Google Scholar 

  • Wu W, Zhang Z, Du D (2019) Set function optimization. J Oper Res Soc China 7(2):183–193

    Article  MathSciNet  Google Scholar 

  • Yang R, Xu D, Du D, Xu Y, Yan X (2019) Maximization of constrained non-submodular functions. In: Proceedings of the 25th international computing and combinatorics conference, pp 615–626

  • Yang R, Xu D, Guo L, Zhang D (2019) Sequence submodular maximization meets streaming. In: Proceedings of the 13th international conference on combinatorial optimization and applications, pp 565–575

  • Yang R, Xu D, Jiang Y, Wang Y, Zhang D (2019) Approximating robust parameterized submodular function maximization in large-scales. Asia-Pac J Oper Res 36(4):671–680

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first two authors are supported by National Natural Science Foundation of China (No. 11531014) and the second author is also supported by Beijing Natural Science Foundation Project No. Z200002. The third author is supported by National Natural Science Foundation of China (No. 61772005) and Natural Science Foundation of Fujian Province (No. 2017J01753). The fourth author is supported by National Natural Science Foundation of China (No. 11871081).

Author information

Authors and Affiliations

  1. Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Ruiqi Yang & Dachuan Xu

  2. School of Mathematical Sciences, University of Chinese Academy Sciences, Beijing, 100049, People’s Republic of China

    Ruiqi Yang

  3. Shandong Key Laboratory of Computer Networks, School of Computer Science and Technology, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, People’s Republic of China

    Longkun Guo

  4. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

Authors
  1. Ruiqi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dachuan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Longkun Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dongmei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Longkun Guo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper appeared in Proceedings of the 13th International Conference on Combinatorial Optimization and Applications (COCOA), 2019, pp. 565-575.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, R., Xu, D., Guo, L. et al. Sequence submodular maximization meets streaming. J Comb Optim 41, 43–55 (2021). https://doi.org/10.1007/s10878-020-00662-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-020-00662-5

Keywords