Abstract
In recent years, submodularity has been found in a wide range of connections and applications with different scientific fields. However, many applications in practice do not fully meet the characteristics of diminishing returns. In this paper, we consider the problem of maximizing unconstrained non-negative weakly-monotone non-submodular set function. The generic submodularity ratio \(\gamma \) is a bridge connecting the non-negative monotone functions and the submodular functions, and no longer applicable to the non-monotone functions. We study a class of non-monotone functions, define as the weakly-monotone function, redefine the submodular ratio related to it and name it weakly-monotone submodularity ratio \(\widehat{\gamma }\), propose a deterministic double greedy algorithm, which implements the \(\frac{\widehat{\gamma }}{\widehat{\gamma }+2}\) approximation of the maximizing unconstrained non-negative weakly-monotone function problem. When \(\widehat{\gamma }=1\), the algorithm achieves an approximate guarantee of 1/3, achieving the same ratio as the deterministic algorithm for the unconstrained submodular maximization problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Badanidiyuru, A., Mirzasoleiman, B., Karbasi, A., Krause, A.: Streaming submodular maximization: massive data summarization on the fly. In: 20th International Proceedings on Knowledge Discovery and Data Mining, New York, NY, USA, pp. 671–680. Association for Computing Machinery (2014)
Bian, A.A., Buhmann, J.M., Krause, A., Tschiatschek, S.: Guarantees for Greedy maximization of non-submodular functions with applications. In: 35th International Conference on Machine Learning, NSW, Australia, vol. 70, pp. 498–507. Proceedings of Machine Learning Research (2017)
Buchbinder, N., Feldman, M., Naor, J.S., Schwartz, R.: A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM J. Comput. 44(5), 1384–1402 (2015)
Buchbinder, N., Feldman, M.: Deterministic algorithms for submodular maximization problems. In: 27th International Symposium on Discrete Algorithms, Arlington, VA, USA, pp. 392–403. Society for Industrial and Applied Mathematics (2016)
Cherenin, V.: Solving some combinatorial problems of optimal planning by the method of successive calculations. In: Conference of Experiences and Perspectives of the Applications of Mathematical Methods and Electronic Computers in Planning, Russian, Mimeograph, Novosibirsk (1962)
Coelho, V.N., et al.: Generic Pareto local search metaheuristic for optimization of targeted offers in a bi-objective direct marketing campaign. Comput. Oper. Res. 78, 578–587 (2017)
Das, A., Kempe, D.: Submodular meets spectral: Greedy algorithms for subset selection, sparse approximation and dictionary selection. In: 28th International Conference on Machine Learning, Bellevue, Washington, USA, pp. 1057–1064. Omnipress (2011)
Dobzinski, S., Vondrak, J.: From query complexity to computational complexity. In: 44th International Symposium on Theory of Computing, New York, NY, USA, pp. 1107–1116. Society for Industrial and Applied Mathematics (2012)
Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)
Feldman, M., Naor, J.S., Schwartz, R.: Nonmonotone submodular maximization via a structural continuous Greedy algorithm. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 342–353. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22006-7_29
Galbiati, G., Maffioli, F.: Approximation algorithms for maximum cut with limited unbalance. Theor. Comput. Sci. 385(1), 78–87 (2007)
Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: 22nd International Symposium on Discrete Algorithms, San Francisco, California, USA, pp. 1098–1116. Society for Industrial and Applied Mathematics (2011)
Gong, S., Nong, Q., Liu, W., Fang, Q.: Parametric monotone function maximization with matroid constraints. J. Global Optim. 75(3), 833–849 (2019). https://doi.org/10.1007/s10898-019-00800-2
Halperin, E., Zwick, U.: Combinatorial approximation algorithms for the maximum directed cut problem. In: 20th International Symposium on Discrete Algorithms, Washington, DC, USA, pp. 1–7. Association for Computing Machinery/Society for Industrial and Applied Mathematics (2001)
Hartline, J., Mirrokni, V., Sundararajan, M.: Optimal marketing strategies over social networks. In: 17th International Proceedings on World Wide Web, Beijing, China, pp. 189–198. Association for Computing Machinery (2008)
Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. Assoc. Comput. Mach. 48(4), 761–777 (2001)
Mansell, R., Collins, B.S.: Introduction: Trust and Crime in Information Societies. Edward Elgar (2005)
Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Facility location and supply chain management - a review. Eur. J. Oper. Res. 196(2), 401–412 (2009)
Mossel, E., Roch, S.: On the submodularity of influence in social networks. In: 30th International Symposium on Theory of Computing, San Diego, California, USA, pp. 128–134. Association for Computing Machinery (2007)
Roughgarden, T., Wang, J.: An optimal learning algorithm for online unconstrained submodular maximization. In: 31st International Conference on Learning Theory, Stockholm, Sweden, pp. 1307–1325. Proceedings of Machine Learning Research (2018)
Tran, A.K., Piran, M.J., Pham, C.: SDN controller placement in IoT networks: an optimized submodularity-based approach. Sensors 19(24), 5474 (2019)
Tsai, Y.-C., Tseng, K.-S.: Deep compressed sensing for learning submodular functions. Sensors 20(9), 2591 (2020)
Tsiang, S.C.: A note on speculation and income stability. Economica 10(40), 27–47 (1943)
Zhang, H., Vorobeychik, Y.: Submodular optimization with routing constraints. In: 30th International Proceedings on Artificial Intelligence, Phoenix, Arizona, USA, pp. 819–825. AAAI Press (2016)
Acknowledgements
The first and third authors are supported by National Natural Science Foundation of China (No. 12131003) and Beijing Natural Science Foundation Project No. Z200002. The second author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant 06446, and Natural Science Foundation of China (Nos. 11771386, 11728104). The fourth author is supported by the Fundamental Research Funds for the Central Universities (No. E1E40108).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Cui, M., Du, D., Xu, D., Yang, R. (2021). Approximation Algorithm for Maximizing Nonnegative Weakly Monotonic Set Functions. In: Mohaisen, D., Jin, R. (eds) Computational Data and Social Networks. CSoNet 2021. Lecture Notes in Computer Science(), vol 13116. Springer, Cham. https://doi.org/10.1007/978-3-030-91434-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-91434-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-91433-2
Online ISBN: 978-3-030-91434-9
eBook Packages: Computer ScienceComputer Science (R0)