Document Open Access Logo

Subset Sum in the Absence of Concentration

Authors Per Austrin, Petteri Kaski, Mikko Koivisto, Jesper Nederlof

PDF
Thumbnail PDF

File

LIPIcs.STACS.2015.48.pdf
  • Filesize: 0.67 MB
  • 14 pages

Document Identifiers

Author Details

Per Austrin
Petteri Kaski
Mikko Koivisto
Jesper Nederlof

Cite As Get BibTex

Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Subset Sum in the Absence of Concentration. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 48-61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.STACS.2015.48

Abstract

We study the exact time complexity of the Subset Sum problem. Our focus is on instances that lack additive structure in the sense that the sums one can form from the subsets of the given integers are not strongly concentrated on any particular integer value. We present a randomized algorithm that runs in O(2^0.3399nB^4) time on instances with the property that no value can arise as a sum of more than B different subsets of the n given integers.

Subject Classification

Keywords
  • subset sum
  • additive combinatorics
  • exponential-time algorithm
  • homomorphic hashing
  • Littlewood--Offord problem

Metrics

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

References

  1. Dimitris Achlioptas. Random satisfiability. In Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors, Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, pages 245-270. IOS Press, 2009. Google Scholar
  2. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jussi Määttä. Space-time tradeoffs for Subset Sum: An improved worst case algorithm. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, ICALP (1), volume 7965 of Lecture Notes in Computer Science, pages 45-56. Springer, 2013. Google Scholar
  3. Anja Becker. The Representation Technique: Application to Hard Problems in Cryptography. PhD thesis, Université de Versailles Saint-Quentin-en-Yvelines, 2012. Google Scholar
  4. Anja Becker, Jean-Sébastien Coron, and Antoine Joux. Improved generic algorithms for hard knapsacks. In Kenneth G. Paterson, editor, EUROCRYPT, volume 6632 of Lecture Notes in Computer Science, pages 364-385. Springer, 2011. Google Scholar
  5. Richard Bellman. Dynamic Programming. Princeton University Press, Princeton, N. J., 1957. Google Scholar
  6. Khodakhast Bibak. Additive combinatorics: With a view towards computer science and cryptography - an exposition. In Jonathan M. Borwein, Igor Shparlinski, and Wadim Zudilin, editors, Number Theory and Related Fields, pages 99-128. Springer, 2013. Google Scholar
  7. Matthijs J. Coster, Antoine Joux, Brian A. Lamacchia, Andrew M. Odlyzko, Claus-Peter Schnorr, and Jacques Stern. Improved low-density subset sum algorithms. Computational Complexity, 2:111-128, 1992. Google Scholar
  8. Itai Dinur, Orr Dunkelman, Nathan Keller, and Adi Shamir. Efficient dissection of composite problems, with applications to cryptanalysis, knapsacks, and combinatorial search problems. In Reihaneh Safavi-Naini and Ran Canetti, editors, CRYPTO, volume 7417 of Lecture Notes in Computer Science, pages 719-740. Springer, 2012. Google Scholar
  9. Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond. Parameterizing by the number of numbers. Theory Comput. Syst., 50(4):675-693, 2012. Google Scholar
  10. Abraham D. Flaxman and Bartosz Przydatek. Solving medium-density subset sum problems in expected polynomial time. In Volker Diekert and Bruno Durand, editors, STACS 2005, volume 3404 of Lecture Notes in Computer Science, pages 305-314. Springer Berlin Heidelberg, 2005. Google Scholar
  11. Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2010. Google Scholar
  12. Godfrey H. Hardy and Edward M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, Oxford, sixth edition, 2008. Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles. Google Scholar
  13. Ellis Horowitz and Sartaj Sahni. Computing partitions with applications to the knapsack problem. J. Assoc. Comput. Mach., 21:277-292, 1974. Google Scholar
  14. Nick Howgrave-Graham and Antoine Joux. New generic algorithms for hard knapsacks. In Henri Gilbert, editor, EUROCRYPT, volume 6110 of Lecture Notes in Computer Science, pages 235-256. Springer, 2010. Google Scholar
  15. Russell Impagliazzo and Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology, 9(4):199-216, 1996. Google Scholar
  16. Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Homomorphic hashing for sparse coefficient extraction. In Dimitrios M. Thilikos and Gerhard J. Woeginger, editors, IPEC, volume 7535 of Lecture Notes in Computer Science, pages 147-158. Springer, 2012. Google Scholar
  17. Jeffrey C. Lagarias and Andrew M. Odlyzko. Solving low-density subset sum problems. J. ACM, 32(1):229-246, 1985. Google Scholar
  18. Daniel Lokshtanov and Jesper Nederlof. Saving space by algebraization. In Leonard J. Schulman, editor, STOC, pages 321-330. ACM, 2010. Google Scholar
  19. Phong Q. Nguyen, Igor E. Shparlinski, and Jacques Stern. Distribution of modular sums and the security of the server aided exponentiation. In Kwok-Yan Lam, Igor Shparlinski, Huaxlong Wang, and Chaoping Xing, editors, Cryptography and Computational Number Theory, volume 20 of Progress in Computer Science and Applied Logic, pages 331-342. Birkhäuser, 2001. Google Scholar
  20. Herbert Robbins. A remark on Stirling’s formula. The American Mathematical Monthly, 62(1):26-29, 1955. Google Scholar
  21. Richard Schroeppel and Adi Shamir. A T = O(2^n/2), S = O(2^n/4) algorithm for certain NP-complete problems. SIAM J. Comput., 10(3):456-464, 1981. Google Scholar
  22. Terence Tao. The dichotomy between structure and randomness, arithmetic progressions, and the primes. In International Congress of Mathematicians. Vol. I, pages 581-608. Eur. Math. Soc., Zürich, 2007. Google Scholar
  23. Terence Tao. Structure and randomness in combinatorics. In FOCS, pages 3-15. IEEE Computer Society, 2007. Google Scholar
  24. Terence Tao. Structure and Randomness. American Mathematical Society, Providence, RI, 2008. Google Scholar
  25. Terence Tao and Van Vu. Additive Combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006. Google Scholar
  26. Terence Tao and Van Vu. A sharp inverse Littlewood-Offord theorem. Random Structures Algorithms, 37(4):525-539, 2010. Google Scholar
  27. Luca Trevisan. Pseudorandomness in computer science and in additive combinatorics. In An irregular mind, volume 21 of Bolyai Soc. Math. Stud., pages 619-650. János Bolyai Math. Soc., Budapest, 2010. Google Scholar
  28. Gerhard J. Woeginger. Exact algorithms for NP-hard problems: A survey. In Michael Jünger, Gerhard Reinelt, and Giovanni Rinaldi, editors, Combinatorial Optimization, volume 2570 of Lecture Notes in Computer Science, pages 185-208. Springer, 2001. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact