Document Open Access Logo

Equal-Subset-Sum Faster Than the Meet-in-the-Middle

Authors Marcin Mucha , Jesper Nederlof , Jakub Pawlewicz , Karol Węgrzycki



PDF
Thumbnail PDF

File

LIPIcs.ESA.2019.73.pdf
  • Filesize: 0.51 MB
  • 16 pages

Document Identifiers

Author Details

Marcin Mucha
  • Institute of Informatics, University of Warsaw, Poland
Jesper Nederlof
  • Eindhoven University of Technology, The Netherlands
Jakub Pawlewicz
  • Institute of Informatics, University of Warsaw, Poland
Karol Węgrzycki
  • Institute of Informatics, University of Warsaw, Poland

Acknowledgements

The authors would like to thank anonymous reviewers for their remarks and suggestions. This research has been initiated during Parameterized Algorithms Retreat of University of Warsaw 2019, Karpacz, 25.02-01.03.2019.

Cite AsGet BibTex

Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki. Equal-Subset-Sum Faster Than the Meet-in-the-Middle. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 73:1-73:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.73

Abstract

In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Equal-Subset-Sum
  • Subset-Sum
  • meet-in-the-middle
  • enumeration technique
  • randomized algorithm

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. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 41-57. SIAM, 2019. 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, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, volume 7965 of Lecture Notes in Computer Science, pages 45-56. Springer, 2013. Google Scholar
  3. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Subset Sum in the Absence of Concentration. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 48-61. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. Google Scholar
  4. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Dense Subset Sum May Be the Hardest. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, volume 47 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  5. Frank Ban, Kamal Jain, Christos H. Papadimitriou, Christos-Alexandros Psomas, and Aviad Rubinstein. Reductions in PPP. Inf. Process. Lett., 145:48-52, 2019. Google Scholar
  6. Nikhil Bansal, Shashwat Garg, Jesper Nederlof, and Nikhil Vyas. Faster space-efficient algorithms for subset sum and k-sum. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 198-209. ACM, 2017. Google Scholar
  7. Cristina Bazgan, Miklos Santha, and Zs. Tuza. Efficient approximation algorithms for the Subset-Sums Equality problem. In International Colloquium on Automata, Languages, and Programming, pages 387-396. Springer, 1998. Google Scholar
  8. Paul Beame, Raphaël Clifford, and Widad Machmouchi. Element Distinctness, Frequency Moments, and Sliding Windows. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 290-299. IEEE Computer Society, 2013. Google Scholar
  9. Anja Becker, Jean-Sébastien Coron, and Antoine Joux. Improved Generic Algorithms for Hard Knapsacks. In Kenneth G. Paterson, editor, Advances in Cryptology - EUROCRYPT 2011 - 30th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tallinn, Estonia, May 15-19, 2011. Proceedings, volume 6632 of Lecture Notes in Computer Science, pages 364-385. Springer, 2011. Google Scholar
  10. Tom Bohman. A sum packing problem of Erdős and the Conway-Guy sequence. Proceedings of the American Mathematical Society, 124(12):3627-3636, 1996. Google Scholar
  11. Karl Bringmann. A Near-linear Pseudopolynomial Time Algorithm for Subset Sum. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 1073-1084, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics. Google Scholar
  12. Mark Cieliebak. Algorithms and hardness results for DNA physical mapping, protein identification, and related combinatorial problems. PhD thesis, ETH Zürich, 2003. Google Scholar
  13. Mark Cieliebak, Stephan Eidenbenz, and Aris Pagourtzis. Composing equipotent teams. In International Symposium on Fundamentals of Computation Theory, pages 98-108. Springer, 2003. Google Scholar
  14. Mark Cieliebak, Stephan Eidenbenz, Aris Pagourtzis, and Konrad Schlude. On the Complexity of Variations of Equal Sum Subsets. Nord. J. Comput., 14(3):151-172, 2008. Google Scholar
  15. Mark Cieliebak, Stephan Eidenbenz, and Paolo Penna. Noisy data make the partial digest problem NP-hard. In International Workshop on Algorithms in Bioinformatics, pages 111-123. Springer, 2003. Google Scholar
  16. John H Conway and Richard K Guy. Sets of natural numbers with distinct subset sums. Notices Amer. Math. Soc, 15:345, 1968. Google Scholar
  17. 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
  18. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On Problems as Hard as CNF-SAT. In Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26-29, 2012, pages 74-84. IEEE Computer Society, 2012. Google Scholar
  19. Paul Erdős. Problems and results in additive number theory. Journal London Wash. Soc, 16:212-215, 1941. Google Scholar
  20. Paul Erdős. A survey of problems in combinatorial number theory. Annals of Discrete Mathematics, 6:89-115, 1980. Google Scholar
  21. Godfrey Harold Hardy, Edward Maitland Wright, et al. An introduction to the theory of numbers. Oxford university press, 1979. Google Scholar
  22. Danny Harnik and Moni Naor. On the Compressibility of NP Instances and Cryptographic Applications. SIAM J. Comput., 39(5):1667-1713, 2010. Google Scholar
  23. Rebecca Hoberg, Harishchandra Ramadas, Thomas Rothvoss, and Xin Yang. Number Balancing is as Hard as Minkowski’s Theorem and Shortest Vector. In Friedrich Eisenbrand and Jochen Könemann, editors, Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, volume 10328 of Lecture Notes in Computer Science, pages 254-266. Springer, 2017. Google Scholar
  24. Ellis Horowitz and Sartaj Sahni. Computing Partitions with Applications to the Knapsack Problem. J. ACM, 21(2):277-292, 1974. Google Scholar
  25. Nick Howgrave-Graham and Antoine Joux. New Generic Algorithms for Hard Knapsacks. In Henri Gilbert, editor, Advances in Cryptology - EUROCRYPT 2010, 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Monaco / French Riviera, May 30 - June 3, 2010. Proceedings, volume 6110 of Lecture Notes in Computer Science, pages 235-256. Springer, 2010. Google Scholar
  26. Russell Impagliazzo and Moni Naor. Efficient Cryptographic Schemes Provably as Secure as Subset Sum. J. Cryptology, 9(4):199-216, 1996. Google Scholar
  27. Ce Jin and Hongxun Wu. A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum. In Jeremy T. Fineman and Michael Mitzenmacher, editors, 2nd Symposium on Simplicity in Algorithms, SOSA@SODA 2019, January 8-9, 2019 - San Diego, CA, USA, volume 69 of OASICS, pages 17:1-17:6. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  28. 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, Chicago, Illinois, USA, 3-5 November 1982, pages 312-320. IEEE Computer Society, 1982. Google Scholar
  29. Konstantinos Koiliaris and Chao Xu. A Faster Pseudopolynomial Time Algorithm for Subset Sum. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 1062-1072, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics. Google Scholar
  30. Konstantinos Koiliaris and Chao Xu. Subset Sum Made Simple. CoRR, abs/1807.08248, 2018. URL: http://arxiv.org/abs/1807.08248.
  31. J. C. Lagarias and Andrew M. Odlyzko. Solving Low-Density Subset Sum Problems. J. ACM, 32(1):229-246, 1985. Google Scholar
  32. Daniel Lokshtanov and Jesper Nederlof. Saving space by algebraization. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 321-330. ACM, 2010. Google Scholar
  33. W Fred Lunnon. Integer sets with distinct subset-sums. Mathematics of Computation, 50(181):297-320, 1988. Google Scholar
  34. Ralph C. Merkle and Martin E. Hellman. Hiding information and signatures in trapdoor knapsacks. IEEE Trans. Information Theory, 24(5):525-530, 1978. Google Scholar
  35. Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Wegrzycki. Equal-Subset-Sum Faster Than the Meet-in-the-Middle. CoRR, abs/1905.02424, 2019. URL: http://arxiv.org/abs/1905.02424.
  36. Jesper Nederlof, Erik Jan van Leeuwen, and Ruben van der Zwaan. Reducing a Target Interval to a Few Exact Queries. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, volume 7464 of Lecture Notes in Computer Science, pages 718-727. Springer, 2012. Google Scholar
  37. Christos H. Papadimitriou. On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. Google Scholar
  38. Herbert Robbins. A remark on Stirling’s formula. The American mathematical monthly, 62(1):26-29, 1955. Google Scholar
  39. Richard Schroeppel and Adi Shamir. A T=O(2^n/2). SIAM J. Comput., 10(3):456-464, 1981. Google Scholar
  40. Adi Shamir. A polynomial-time algorithm for breaking the basic Merkle-Hellman cryptosystem. IEEE Trans. Information Theory, 30(5):699-704, 1984. Google Scholar
  41. Katerina Sotiraki, Manolis Zampetakis, and Giorgos Zirdelis. PPP-Completeness with Connections to Cryptography. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 148-158. IEEE Computer Society, 2018. Google Scholar
  42. David A. Wagner. A Generalized Birthday Problem. In Moti Yung, editor, Advances in Cryptology - CRYPTO 2002, 22nd Annual International Cryptology Conference, Santa Barbara, California, USA, August 18-22, 2002, Proceedings, volume 2442 of Lecture Notes in Computer Science, pages 288-303. Springer, 2002. Google Scholar
  43. Gerhard J. Woeginger. Space and Time Complexity of Exact Algorithms: Some Open Problems (Invited Talk). In Rodney G. Downey, Michael R. Fellows, and Frank K. H. A. Dehne, editors, Parameterized and Exact Computation, First International Workshop, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Proceedings, volume 3162 of Lecture Notes in Computer Science, pages 281-290. Springer, 2004. Google Scholar
  44. Gerhard J. Woeginger. Open problems around exact algorithms. Discrete Applied Mathematics, 156(3):397-405, 2008. Google Scholar
  45. Gerhard J. Woeginger and Zhongliang Yu. On the Equal-Subset-Sum Problem. Inf. Process. Lett., 42(6):299-302, 1992. 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