Document Open Access Logo

Simple Deterministic Approximation for Submodular Multiple Knapsack Problem

Authors Xiaoming Sun, Jialin Zhang, Zhijie Zhang

PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.98.pdf
  • Filesize: 0.73 MB
  • 15 pages

Document Identifiers

Author Details

Xiaoming Sun
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Jialin Zhang
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Zhijie Zhang
  • Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University, China

Funding

This work was supported in part by the National Natural Science Foundation of China Grants No. 61832003, 62272441.

Cite AsGet BibTex

Xiaoming Sun, Jialin Zhang, and Zhijie Zhang. Simple Deterministic Approximation for Submodular Multiple Knapsack Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 98:1-98:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.98

Abstract

Submodular maximization has been a central topic in theoretical computer science and combinatorial optimization over the last decades. Plenty of well-performed approximation algorithms have been designed for the problem over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP). In SMKP, the profits of each subset of elements are specified by a monotone submodular function. The goal is to find a feasible packing of elements over multiple bins (knapsacks) to maximize the profit. Recently, Fairstein et al. [ESA20] proposed a nearly optimal (1-e^{-1}-ε)-approximation algorithm for SMKP. Their algorithm is obtained by combining configuration LP, a grouping technique for bin packing, and the continuous greedy algorithm for submodular maximization. As a result, the algorithm is somewhat sophisticated and inherently randomized. In this paper, we present an arguably simple deterministic combinatorial algorithm for SMKP, which achieves a (1-e^{-1}-ε)-approximation ratio. Our algorithm is based on very different ideas compared with Fairstein et al. [ESA20].

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Submodular maximization
  • knapsack problem
  • deterministic algorithm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1497-1514, 2014. URL: https://doi.org/10.1137/1.9781611973402.110.
  2. Niv Buchbinder and Moran Feldman. Deterministic algorithms for submodular maximization problems. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 392-403, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch29.
  3. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 649-658, 2012. URL: https://doi.org/10.1109/FOCS.2012.73.
  4. Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. Submodular maximization with cardinality constraints. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1433-1452, 2014. URL: https://doi.org/10.1137/1.9781611973402.106.
  5. Chandra Chekuri and Sanjeev Khanna. A PTAS for the multiple knapsack problem. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA., pages 213-222, 2000. URL: http://dl.acm.org/citation.cfm?id=338219.338254.
  6. Wenceslas Fernandez de la Vega and George S. Lueker. Bin packing can be solved within 1+epsilon in linear time. Combinatorica, 1(4):349-355, 1981. URL: https://doi.org/10.1007/BF02579456.
  7. Alina Ene and Huy L. Nguyen. Constrained submodular maximization: Beyond 1/e. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 248-257, 2016. URL: https://doi.org/10.1109/FOCS.2016.34.
  8. Alina Ene and Huy L. Nguyen. A nearly-linear time algorithm for submodular maximization with a knapsack constraint. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece., pages 53:1-53:12, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.53.
  9. Yaron Fairstein, Ariel Kulik, Joseph (Seffi) Naor, Danny Raz, and Hadas Shachnai. A (1-e^-1-ε)-approximation for the monotone submodular multiple knapsack problem. In 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 44:1-44:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.44.
  10. Yaron Fairstein, Ariel Kulik, and Hadas Shachnai. Modular and submodular optimization with multiple knapsack constraints via fractional grouping. In 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 41:1-41:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.41.
  11. Moran Feldman. Maximization problems with submodular objective functions. Technion-Israel Institute of Technology, Faculty of Computer Science, 2013. Google Scholar
  12. Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 570-579, 2011. URL: https://doi.org/10.1109/FOCS.2011.46.
  13. Moran Feldman, Zeev Nutov, and Elad Shoham. Practical budgeted submodular maximization. Algorithmica, 85(5):1332-1371, 2023. URL: https://doi.org/10.1007/s00453-022-01071-2.
  14. Yuval Filmus and Justin Ward. A tight combinatorial algorithm for submodular maximization subject to a matroid constraint. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 659-668, 2012. URL: https://doi.org/10.1109/FOCS.2012.55.
  15. Marshall L. Fisher, George L. Nemhauser, and Laurence A. Wolsey. An analysis of approximations for maximizing submodular set functions - II. In Polyhedral combinatorics, pages 73-87. Springer, 1978. Google Scholar
  16. Shayan Oveis Gharan and Jan Vondrák. Submodular maximization by simulated annealing. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1098-1116, 2011. URL: https://doi.org/10.1137/1.9781611973082.83.
  17. Klaus Jansen. Parameterized approximation scheme for the multiple knapsack problem. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 665-674, 2009. URL: https://doi.org/10.1137/1.9781611973068.73.
  18. Hans Kellerer. A polynomial time approximation scheme for the multiple knapsack problem. In Randomization, Approximation, and Combinatorial Algorithms and Techniques, RANDOM-APPROX'99, Berkeley, CA, USA, August 8-11, 1999, Proceedings, pages 51-62, 1999. URL: https://doi.org/10.1007/978-3-540-48413-4_6.
  19. David Kempe, Jon M. 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, Washington, DC, USA, August 24 - 27, 2003, pages 137-146. ACM, 2003. URL: https://doi.org/10.1145/956750.956769.
  20. Ariel Kulik, Roy Schwartz, and Hadas Shachnai. A refined analysis of submodular greedy. Oper. Res. Lett., 49(4):507-514, 2021. URL: https://doi.org/10.1016/j.orl.2021.04.006.
  21. Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. Non-monotone submodular maximization under matroid and knapsack constraints. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 323-332. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536459.
  22. Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res., 35(4):795-806, 2010. URL: https://doi.org/10.1287/moor.1100.0463.
  23. George L. Nemhauser and Laurence A. Wolsey. Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res., 3(3):177-188, 1978. URL: https://doi.org/10.1287/moor.3.3.177.
  24. George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Program., 14(1):265-294, 1978. URL: https://doi.org/10.1007/BF01588971.
  25. Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett., 32(1):41-43, 2004. URL: https://doi.org/10.1016/S0167-6377(03)00062-2.
  26. Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 67-74, 2008. URL: https://doi.org/10.1145/1374376.1374389.
  27. Jan Vondrák, Chandra Chekuri, and Rico Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 783-792, 2011. URL: https://doi.org/10.1145/1993636.1993740.
  28. Yuichi Yoshida. Maximizing a monotone submodular function with a bounded curvature under a knapsack constraint. SIAM J. Discret. Math., 33(3):1452-1471, 2019. URL: https://doi.org/10.1137/16M1107644.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact