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Nonsubmodular Optimization

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Nonlinear Combinatorial Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 147))

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Abstract

The nonsubmodular optimization is a hot research topic in the study of nonlinear combinatorial optimizations. We discuss several approaches to deal with such optimization problems, including supermodular degree, curvature, algorithms based on DS decomposition, and sandwich method.

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References

  1. Bai, W., Bilmes, J.A.: Greed is still good: maximizing monotone submodular+ supermodular functions (2018). Preprint. arXiv:1801.07413

    Google Scholar 

  2. Barhan, A., Shakhomirov, A.: Methods for sentiment analysis of twitter messages. In: 12th Conference of FRUCT Association (2012)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, W., Lin, T., Tan, Z., Zhao, M., Zhou, X.: Robust influence maximization. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 795–804. ACM, New York (2016)

    Google Scholar 

  5. Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Netw. Int. J. 42(4), 202–208 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Ding, L., Gao, X., Wu, W., Lee, W., Zhu, X., Du, D.-Z.: Distributed construction of connected dominating sets with minimum routing cost in wireless networks. In: 2010 IEEE 30th International Conference on Distributed Computing Systems (ICDCS), pp. 448–457. IEEE, Piscataway (2010)

    Google Scholar 

  7. Ding, L., Wu, W., Willson, J.K., Du, H., Lee, W.: Construction of directional virtual backbones with minimum routing cost in wireless networks. In: INFOCOM, 2011 Proceedings IEEE, pp. 1557–1565. IEEE, Piscataway (2011)

    Google Scholar 

  8. Du, H., Wu, W., Lee, W., Liu, Q., Zhang, Z., Du, D.-Z.: On minimum submodular cover with submodular cost. J. Glob. Optim. 50(2), 229–234 (2011)

    Article  MathSciNet  Google Scholar 

  9. Du, H., Ye, Q., Wu, W., Lee, W., Li, D., Du, D., Howard, S.: Constant approximation for virtual backbone construction with guaranteed routing cost in wireless sensor networks. In: INFOCOM, 2011 Proceedings IEEE, pp. 1737–1744. IEEE, Piscataway (2011)

    Google Scholar 

  10. Du, H., Ye, Q., Zhong, J., Wang, Y., Lee, W., Park, H.: Polynomial-time approximation scheme for minimum connected dominating set under routing cost constraint in wireless sensor networks. Theor. Comput. Sci. 447, 38–43 (2012)

    Article  MathSciNet  Google Scholar 

  11. Du, H., Wu, W., Ye, Q., Li, D., Lee, W., Xu, X.: CDS-based virtual backbone construction with guaranteed routing cost in wireless sensor networks. IEEE Trans. Parallel Distrib. Syst. 24(4), 652–661 (2013)

    Article  Google Scholar 

  12. Edmonds, J., Pruhs, K.: Scalably scheduling processes with arbitrary speedup curves. In: Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 685–692. SIAM, Philadelphia (2009)

    Google Scholar 

  13. Feige, U., Izsak, R.: Welfare maximization and the supermodular degree. In: Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pp. 247–256. ACM, New York (2013)

    Google Scholar 

  14. Feige, U., Mirrokni, V.S., Vondrak, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)

    Article  MathSciNet  Google Scholar 

  15. Feldman, M., Izsak, R.: Constrained monotone function maximization and the supermodular degree. (2014). Preprint. arXiv:1407.6328

    Google Scholar 

  16. Feldman, M., Izsak, R.: Building a good team: secretary problems and the supermodular degree. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1651–1670. SIAM, Philadelphia (2017)

    Google Scholar 

  17. Feldman, M., Naor, J., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 570–579. IEEE, Piscataway (2011)

    Google Scholar 

  18. Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions—II. In: Polyhedral Combinatorics, pp. 73–87. Springer, Berlin (1978)

    Google Scholar 

  19. Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)

    Article  MathSciNet  Google Scholar 

  20. Guo, J., Wu, W. Viral marketing with complementary products. In: Du, D.-Z., Pardalos, P.M., Zhang, Z. (eds.) Nonlinear Combinatorial Optimization. Springer International Publishing, Cham (2019)

    Google Scholar 

  21. Iyer, R., Bilmes, J.: Algorithms for approximate minimization of the difference between submodular functions, with applications. (2012). Preprint. arXiv:1207.0560

    Google Scholar 

  22. Iyer, R.K., Bilmes, J.A.: Submodular optimization with submodular cover and submodular knapsack constraints. In: Advances in Neural Information Processing Systems, pp. 2436–2444 (2013)

    Google Scholar 

  23. Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Non-monotone submodular maximization under matroid and knapsack constraints. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 323–332. ACM, New York (2009)

    Google Scholar 

  24. Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J., Glance, N.: Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 420–429. ACM, New York (2007)

    Google Scholar 

  25. Lu, W., Chen, W., Lakshmanan, L.V.S.: From competition to complementarity: comparative influence diffusion and maximization. Proc. VLDB Endowment 9(2), 60–71 (2015)

    Article  Google Scholar 

  26. Maehara, T., Murota, K.: A framework of discrete dc programming by discrete convex analysis. Math. Program. 152(1–2), 435–466 (2015)

    Article  MathSciNet  Google Scholar 

  27. Narasimhan, M., Bilmes, J.A.: A submodular-supermodular procedure with applications to discriminative structure learning (2012). Preprint. arXiv:1207.1404

    Google Scholar 

  28. Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions I. Math. Program. 14(1), 265–294 (1978)

    Article  MathSciNet  Google Scholar 

  29. Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Program. 118(2), 237–251 (2009)

    Article  MathSciNet  Google Scholar 

  30. Ruan, L., Du, H., Jia, X., Wu, W., Li, Y., Ko, K.-I.: A greedy approximation for minimum connected dominating sets. Theor. Comput. Sci. 329(1–3), 325–330 (2004)

    Article  MathSciNet  Google Scholar 

  31. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory Ser. B 80(2), 346–355 (2000)

    Article  MathSciNet  Google Scholar 

  32. Sivakumar, R., Das, B., Bharghavan, V.: An improved spine-based infrastructure for routing in ad hoc networks. In: IEEE Symposium on Computers and Communications, vol., 98 (1998)

    Google Scholar 

  33. Stojmenovic, I., Seddigh, M., Zunic, J.: Dominating sets and neighbor elimination-based broadcasting algorithms in wireless networks. IEEE Trans. Parallel Distrib. Syst. 13(1), 14–25 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Svitkina, Z., Fleischer, L.: Submodular approximation: sampling-based algorithms and lower bounds. SIAM J. Comput. 40(6), 1715–1737 (2011)

    Article  MathSciNet  Google Scholar 

  36. Tong, A., Du, D.-Z., Wu, W.: On misinformation containment in online social networks. In: Advances in Neural Information Processing Systems, pp. 339–349 (2018)

    Google Scholar 

  37. Wan, P.-J., Alzoubi, K.M., Frieder, O.: Distributed construction of connected dominating set in wireless ad hoc networks. In: INFOCOM 2002. Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE, vol. 3, pp. 1597–1604. IEEE, Piscataway (2002)

    Google Scholar 

  38. Wan, P.-J., Du, D.-Z., Pardalos, P., Wu, W.: Greedy approximations for minimum submodular cover with submodular cost. Comput. Optim. Appl. 45(2), 463–474 (2010)

    Article  MathSciNet  Google Scholar 

  39. Wang, Z., Yang, Y., Pei, J., Chu, L., Chen, E.: Activity maximization by effective information diffusion in social networks. IEEE Trans. Knowl. Data Eng. 29(11), 2374–2387 (2017)

    Article  Google Scholar 

  40. Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)

    Article  MathSciNet  Google Scholar 

  41. Wu, J., Li, H.: On calculating connected dominating set for efficient routing in ad hoc wireless networks. In: Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 7–14. ACM, New York (1999)

    Google Scholar 

  42. Wu, B., Lyu, S., Ghanem, B.: Constrained submodular minimization for missing labels and class imbalance in multi-label learning. In: AAAI, pp. 2229–2236 (2016)

    Google Scholar 

  43. Wu, C., Wang, Y., Lu, Z., Pardalos, P.M., Xu, D., Zhang, Z., Du, D.-Z.: Solving the degree-concentrated fault-tolerant spanning subgraph problem by DC programming. Math. Program. 169(1), 255–275 (2018)

    Article  MathSciNet  Google Scholar 

  44. Wu, W., Zhang, Z., Du, D.Z.: Set function optimization. J. Oper. Res. China (to appear)

    Google Scholar 

  45. Zhu, J., Zhu, J., Ghosh, S., Wu, W., Yuan, J.: Social influence maximization in hypergraph in social networks. IEEE Trans. Netw. Sci. Eng. (2018)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA

    Weili Wu & Ding-Zhu Du

  2. Department of Computer Science, Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

Authors
  1. Weili Wu
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  2. Zhao Zhang
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  3. Ding-Zhu Du
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Corresponding author

Correspondence to Ding-Zhu Du .

Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Department of Industrial & Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

  3. Department of Computer Science, Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

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Wu, W., Zhang, Z., Du, DZ. (2019). Nonsubmodular Optimization. In: Du, DZ., Pardalos, P., Zhang, Z. (eds) Nonlinear Combinatorial Optimization. Springer Optimization and Its Applications, vol 147. Springer, Cham. https://doi.org/10.1007/978-3-030-16194-1_6

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