No Polynomial Kernels for Knapsack

Authors Klaus Heeger , Danny Hermelin , Matthias Mnich , Dvir Shabtay



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.83.pdf
  • Filesize: 0.79 MB
  • 17 pages

Document Identifiers

Author Details

Klaus Heeger
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Danny Hermelin
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Matthias Mnich
  • Institute for Algorithms and Complexity, Hamburg University of Technology, Hamburg, Germany
Dvir Shabtay
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Cite As Get BibTex

Klaus Heeger, Danny Hermelin, Matthias Mnich, and Dvir Shabtay. No Polynomial Kernels for Knapsack. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 83:1-83:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.83

Abstract

This paper focuses on kernelization algorithms for the fundamental Knapsack problem. A kernelization algorithm (or kernel) is a polynomial-time reduction from a problem onto itself, where the output size is bounded by a function of some problem-specific parameter. Such algorithms provide a theoretical model for data reduction and preprocessing and are central in the area of parameterized complexity. In this way, a kernel for Knapsack for some parameter k reduces any instance of Knapsack to an equivalent instance of size at most f(k) in polynomial time, for some computable function f. When f(k) = k^{O(1)} then we call such a reduction a polynomial kernel. 
Our study focuses on two natural parameters for Knapsack: The number w_# of different item weights, and the number p_# of different item profits. Our main technical contribution is a proof showing that Knapsack does not admit a polynomial kernel for any of these two parameters under standard complexity-theoretic assumptions. Our proof discovers an elaborate application of the standard kernelization lower bound framework, and develops along the way novel ideas that should be useful for other problems as well. We complement our lower bounds by showing that Knapsack admits a polynomial kernel for the combined parameter w_# ⋅ p_#.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Knapsack
  • polynomial kernels
  • compositions
  • number of different weights
  • number of different profits

Metrics

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

References

  1. Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Seth-based lower bounds for subset sum and bicriteria path. ACM Trans. Algorithms, 18(1):6:1-6:22, 2022. URL: https://doi.org/10.1145/3450524.
  2. Kyriakos Axiotis and Christos Tzamos. Capacitated dynamic programming: Faster knapsack and graph algorithms. In Proc. of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 19:1-19:13, 2019. Google Scholar
  3. MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Saeed Seddighin, and Clifford Stein. Fast algorithms for knapsack via convolution and prediction. In Proc. of the 50th ACM Symposium on the Theory Of Computing (STOC), pages 1269-1282, 2018. Google Scholar
  4. Richard E. Bellman. Dynamic programming. Princeton University Press, 1957. Google Scholar
  5. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. URL: https://doi.org/10.1016/J.JCSS.2009.04.001.
  6. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. URL: https://doi.org/10.1137/120880240.
  7. Ernest F. Brickell and Andrew M. Odlyzko. Cryptanalysis: A survey of recent results. Proceedings of the IEEE, 76(5):578-593, 1988. Google Scholar
  8. Karl Bringmann. Knapsack with small items in near-quadratic time. In Proc. of the 56th ACM Symposium on the Theory of Computing (STOC), 2024. To appear. Google Scholar
  9. Karl Bringmann and Alejandro Cassis. Faster knapsack algorithms via bounded monotone min-plus-convolution. In Proc. of the 49th International Colloquium on Automata, Languages, and Programming, (ICALP), pages 31:1-31:21, 2022. Google Scholar
  10. Karl Bringmann and Alejandro Cassis. Faster 0-1-knapsack via near-convex min-plus-convolution. In Proc. of the 31st Annual European Symposium on Algorithms (ESA), pages 24:1-24:16, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.24.
  11. Harry Buhrman, Bruno Loff, and Leen Torenvliet. Hardness of approximation for knapsack problems. Theory of Computing Systems, 56(2):372-393, 2015. Google Scholar
  12. Timothy M. Chan. Approximation schemes for 0-1 knapsack. In 1st Symposium on Simplicity in Algorithms (SOSA), pages 5:1-5:12, 2018. URL: https://doi.org/10.4230/OASICS.SOSA.2018.5.
  13. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. Faster algorithms for bounded knapsack and bounded subset sum via fine-grained proximity results. In Proc. of the 2024 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4828-4848, 2024. URL: https://doi.org/10.1137/1.9781611977912.171.
  14. Benny Chor and Ronald R. Rivest. A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Transactions on Information Theory, 34(5):901-909, 1988. Google Scholar
  15. William J. Cook, A. M. H. Gerards, Alexander Schrijver, and Éva Tardos. Sensitivity theorems in integer linear programming. Mathematical Programming, 34(3):251-264, 1986. URL: https://doi.org/10.1007/BF01582230.
  16. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  17. Marek Cygan, Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. On problems equivalent to (min, +)-convolution. ACM Transactions on Algorithms, 15(1):14:1-14:25, 2019. Google Scholar
  18. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Journal of the ACM, 61(4):23:1-23:27, 2014. Google Scholar
  19. Mingyang Deng, Ce Jin, and Xiao Mao. Approximating knapsack and partition via dense subset sums. In Proc. of the 2023 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2961-2979, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH113.
  20. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and ids. ACM Transactions on Algorithms, 11(2):13:1-13:20, 2014. Google Scholar
  21. Andrew Drucker. New limits to classical and quantum instance compression. SIAM Journal on Computing, 44(5):1443-1479, 2015. Google Scholar
  22. Michael Etscheid, Stefan Kratsch, Matthias Mnich, and Heiko Röglin. Polynomial kernels for weighted problems. Journal of Computer and System Sciences, 84:1-10, 2017. URL: https://doi.org/10.1016/J.JCSS.2016.06.004.
  23. Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond. Parameterizing by the number of numbers. Theory of Computing Systems, 50(4):675-693, 2012. Google Scholar
  24. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019. Google Scholar
  25. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences, 77(1):91-106, 2011. Google Scholar
  26. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7(1):49-65, 1987. URL: https://doi.org/10.1007/BF02579200.
  27. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985. Google Scholar
  28. Klaus Heeger, Danny Hermelin, Matthias Mnich, and Dvir Shabtay. No polynomial kernels for knapsack. CoRR, 2023. URL: https://doi.org/10.48550/arXiv.2308.12593.
  29. Danny Hermelin, Shlomo Karhi, Michael L. Pinedo, and Dvir Shabtay. New algorithms for minimizing the weighted number of tardy jobs on a single machine. Annals of Operations Research, 298(1):271-287, 2021. Google Scholar
  30. Russell Impagliazzo and Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology, 9(4):199-216, 1996. Google Scholar
  31. Klaus Jansen, Felix Land, and Kati Land. Bounding the running time of algorithms for scheduling and packing problems. SIAM Journal on Discrete Mathematics, 30(1):343-366, 2016. Google Scholar
  32. Ce Jin. An improved FPTAS for 0-1 knapsack. In Proc. of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 76:1-76:14, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.76.
  33. Ce Jin. 0-1 knapsack in nearly quadratic time. In Proc. of the 56th ACM Symposium on the Theory of Computing (STOC), 2024. To appear. Google Scholar
  34. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415-440, 1987. URL: https://doi.org/10.1287/MOOR.12.3.415.
  35. Richard M. Karp. Reducibility among combinatorial problems. In Proc. of a symposium on the Complexity of Computer Computations, pages 85-103. Plenum Press, 1972. Google Scholar
  36. Hans Kellerer and Ulrich Pferschy. Improved dynamic programming in connection with an FPTAS for the knapsack problem. Journal of Combinatorial Optimization, 8(1):5-11, 2004. Google Scholar
  37. Hans Kellerer, Ulrich Pferschy, and David Pisinger. Knapsack problems. Springer, 2004. Google Scholar
  38. Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the fine-grained complexity of one-dimensional dynamic programming. In Proc. of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), pages 21:1-21:15, 2017. URL: https://doi.org/10.4230/LIPICS.ICALP.2017.21.
  39. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  40. Silvano Martello and Paolo Toth. Knapsack problems: algorithms and computer implementations. John Wiley & Sons, Inc., 1990. Google Scholar
  41. Ralph Merkle and Martin Hellman. Hiding information and signatures in trapdoor knapsacks. IEEE Transactions on Information Theory, 24(5):525-530, 1978. Google Scholar
  42. Andrew M. Odlyzko. The rise and fall of knapsack cryptosystems. Cryptology and Computational Number Theory, 42:75-88, 1990. Google Scholar
  43. Michael Pinedo. Scheduling: Theory, Algorithms, and Systems. Springer, 2016. Google Scholar
  44. David Pisinger. Linear time algorithms for knapsack problems with bounded weights. Journal of Algorithms, 32:1-14, 1999. Google Scholar
  45. Adam Polak, Lars Rohwedder, and Karol Węgrzycki. Knapsack and subset sum with small items. In Proc. of the 48th International Colloquium on Automata, Languages, and Programming (ICALP), pages 106:1-106:19, 2021. Google Scholar
  46. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In Proc. of the 64th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 974-988, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  47. Chee-Keng Yap. Some consequences of non-uniform conditions on uniform classes. Theoretical Computer Science, 26:287-300, 1983. URL: https://doi.org/10.1016/0304-3975(83)90020-8.
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