An AFPTAS for Bin Packing with Partition Matroid via a New Method for LP Rounding

Authors Ilan Doron-Arad, Ariel Kulik, Hadas Shachnai



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2023.22.pdf
  • Filesize: 0.93 MB
  • 16 pages

Document Identifiers

Author Details

Ilan Doron-Arad
  • Computer Science Department, Technion, Haifa, Israel
Ariel Kulik
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Hadas Shachnai
  • Computer Science Department, Technion, Haifa, Israel

Cite As Get BibTex

Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. An AFPTAS for Bin Packing with Partition Matroid via a New Method for LP Rounding. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.22

Abstract

We consider the Bin Packing problem with a partition matroid constraint. The input is a set of items of sizes in [0,1], and a partition matroid over the items. The goal is to pack the items in a minimum number of unit-size bins, such that each bin forms an independent set in the matroid. This variant of classic Bin Packing has natural applications in secure storage on the Cloud, as well as in equitable scheduling and clustering with fairness constraints.
Our main result is an asymptotic fully polynomial-time approximation scheme (AFPTAS) for Bin Packing with a partition matroid constraint. This scheme generalizes the known AFPTAS for Bin Packing with Cardinality Constraints and improves the existing asymptotic polynomial-time approximation scheme (APTAS) for Group Bin Packing, which are both special cases of Bin Packing with partition matroid. We derive the scheme via a new method for rounding a (fractional) solution for a configuration-LP. Our method uses this solution to obtain prototypes, in which items are interpreted as placeholders for other items, and applies fractional grouping to modify a fractional solution (prototype) into one having desired integrality properties.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • bin packing
  • partition-matroid
  • AFPTAS
  • LP-rounding

Metrics

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

References

  1. Ron Adany, Moran Feldman, Elad Haramaty, Rohit Khandekar, Baruch Schieber, Roy Schwartz, Hadas Shachnai, and Tami Tamir. All-or-nothing generalized assignment with application to scheduling advertising campaigns. In Integer Programming and Combinatorial Optimization - 16th International Conference, IPCO, pages 13-24, 2013. Google Scholar
  2. Nikhil Bansal, Alberto Caprara, and Maxim Sviridenko. A new approximation method for set covering problems, with applications to multidimensional bin packing. SIAM Journal on Computing, 39(4):1256-1278, 2010. Google Scholar
  3. Nikhil Bansal, Marek Eliáš, and Arindam Khan. Improved approximation for vector bin packing. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, pages 1561-1579. SIAM, 2016. Google Scholar
  4. Alberto Caprara, Hans Kellerer, and Ulrich Pferschy. Approximation schemes for ordered vector packing problems. Naval Research Logistics (NRL), 50(1):58-69, 2003. Google Scholar
  5. Chandra Chekuri and Sanjeev Khanna. A polynomial time approximation scheme for the multiple knapsack problem. SIAM Journal on Computing, 35(3):713-728, 2005. Google Scholar
  6. Syamantak Das and Andreas Wiese. On minimizing the makespan when some jobs cannot be assigned on the same machine. In 25th Annual European Symposium on Algorithms, ESA, pages 31:1-31:14, 2017. Google Scholar
  7. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. An APTAS for bin packing with clique-graph conflicts. arXiv preprint, 2020. URL: https://arxiv.org/abs/2011.04273.
  8. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. An APTAS for bin packing with clique-graph conflicts. In 17th International Symposium on Algorithms and Data Structures, WADS, pages 286-299, 2021. Google Scholar
  9. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. Bin packing with partition matroid can be approximated within o (opt) bins. arXiv preprint, 2022. URL: https://arxiv.org/abs/2212.01025.
  10. Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. An FPTAS for Budgeted Laminar Matroid Independent Set. arXiv preprint, 2023. URL: https://arxiv.org/abs/2304.13984.
  11. Ilan Doron-Arad and Hadas Shachnai. Approximating bin packing with conflict graphs via maximization techniques. Proc. WG, 2023. Google Scholar
  12. Jack Edmonds. Minimum partition of a matroid into independent subsets. J. Res. Nat. Bur. Standards Sect. B, 69:67-72, 1965. Google Scholar
  13. Ibtissam Ennajjar, Youness Tabii, and Abdelhamid Benkaddour. Securing data in cloud computing by classification. In Proceedings of the 2nd international Conference on Big Data, Cloud and Applications, pages 1-5, 2017. Google Scholar
  14. Leah Epstein and Asaf Levin. On bin packing with conflicts. SIAM Journal on Optimization, 19(3):1270-1298, 2008. Google Scholar
  15. Leah Epstein and Asaf Levin. AFPTAS results for common variants of bin packing: A new method for handling the small items. SIAM Journal on Optimization, 20(6):3121-3145, 2010. Google Scholar
  16. 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, pages 41:1-41:16, 2021. Google Scholar
  17. W Fernandez de La Vega and George S. Lueker. Bin packing can be solved within 1+ ε in linear time. Combinatorica, 1(4):349-355, 1981. Google Scholar
  18. Lisa Fleischer, Michel X Goemans, Vahab S Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum separable assignment problems. Mathematics of Operations Research, 36(3):416-431, 2011. Google Scholar
  19. Harold N Gabow and Herbert H Westermann. Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica, 7(1):465-497, 1992. Google Scholar
  20. Michael R Garey and David S Johnson. Computers and intractability. A Guide to the, 1979. Google Scholar
  21. Kilian Grage, Klaus Jansen, and Kim-Manuel Klein. An EPTAS for machine scheduling with bag-constraints. In The 31st ACM Symposium on Parallelism in Algorithms and Architectures, pages 135-144, 2019. Google Scholar
  22. Michael D Grigoriadis, Leonid G Khachiyan, Lorant Porkolab, and Jorge Villavicencio. Approximate max-min resource sharing for structured concave optimization. SIAM Journal on Optimization, 11(4):1081-1091, 2001. Google Scholar
  23. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2. Springer Science & Business Media, 2012. Google Scholar
  24. Marcos Guerine, Murilo B Stockinger, Isabel Rosseti, Luidi G Simonetti, Kary ACS Ocaña, Alexandre Plastino, and Daniel de Oliveira. A provenance-based heuristic for preserving results confidentiality in cloud-based scientific workflows. Future Generation Computer Systems, 97:697-713, 2019. Google Scholar
  25. Klaus Heeger, Danny Hermelin, George B Mertzios, Hendrik Molter, Rolf Niedermeier, and Dvir Shabtay. Equitable scheduling on a single machine. Journal of Scheduling, pages 1-17, 2022. Google Scholar
  26. Rebecca Hoberg and Thomas Rothvoß. A logarithmic additive integrality gap for bin packing. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2616-2625. SIAM, 2017. Google Scholar
  27. Klaus Jansen. An approximation scheme for bin packing with conflicts. Journal of combinatorial optimization, 3(4):363-377, 1999. Google Scholar
  28. Klaus Jansen. Parameterized approximation scheme for the multiple knapsack problem. SIAM Journal on Computing, 39(4):1392-1412, 2010. Google Scholar
  29. Klaus Jansen. A fast approximation scheme for the multiple knapsack problem. In International Conference on Current Trends in Theory and Practice of Computer Science, pages 313-324. Springer, 2012. Google Scholar
  30. Klaus Jansen, Marten Maack, and Malin Rau. Approximation schemes for machine scheduling with resource (in-) dependent processing times. ACM Transactions on Algorithms (TALG), 15(3):1-28, 2019. Google Scholar
  31. Klaus Jansen and Sabine R. Öhring. Approximation algorithms for time constrained scheduling. Inf. Comput., 132(2):85-108, 1997. Google Scholar
  32. Narendra Karmarkar and Richard M. Karp. An efficient approximation scheme for the one-dimensional bin-packing problem. In 23rd Annual Symposium on Foundations of Computer Science, pages 312-320. IEEE, 1982. Google Scholar
  33. Hans Kellerer and Ulrich Pferschy. Cardinality constrained bin-packing problems. Annals of Operations Research, 92:335-348, 1999. Google Scholar
  34. Kenneth L Krause, Vincent Y Shen, and Herbert D Schwetman. Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. Journal of the ACM (JACM), 22(4):522-550, 1975. Google Scholar
  35. KL Krause, Vincent Y Shen, and Herbert D Schwetman. Errata:"analysis of several task-scheduling algorithms for a model of multiprogramming computer systems". Journal of the ACM (JACM), 24(3):527, 1977. Google Scholar
  36. Bill McCloskey and AJ. Shankar. Approaches to bin packing with clique-graph conflicts. Computer Science Division, University of California, 2005. Google Scholar
  37. Y. Oh and S.H. Son. On a constrained bin-packing problem. Technical Report CS-95-14, 1995. Google Scholar
  38. Serge A Plotkin, David B Shmoys, and Éva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20(2):257-301, 1995. Google Scholar
  39. Tai Le Quy, Gunnar Friege, and Eirini Ntoutsi. Multiple fairness and cardinality constraints for students-topics grouping problem. arXiv preprint, 2022. URL: https://arxiv.org/abs/2206.09895.
  40. David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011. Google Scholar
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