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Dense Subset Sum May Be the Hardest

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

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Per Austrin
Petteri Kaski
Mikko Koivisto
Jesper Nederlof

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Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Dense Subset Sum May Be the Hardest. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 13:1-13:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


The SUBSET SUM problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O^*(2^{n/2})-time algorithm for SUBSET SUM by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O^*(2^{(0.5-delta)*n})-time algorithm, with some constant delta > 0. Continuing an earlier work [STACS 2015], we study SUBSET SUM parameterized by the maximum bin size beta, defined as the largest number of subsets of the n input integers that yield the same sum. For every epsilon > 0 we give a truly faster algorithm for instances with beta <= 2^{(0.5-epsilon)*n}, as well as instances with beta >= 2^{0.661n}. Consequently, we also obtain a characterization in terms of the popular density parameter n/log_2(t): if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for SUBSET SUM and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.
  • subset sum
  • additive combinatorics
  • exponential-time algorithm
  • homo-morphic hashing
  • littlewood–offord problem


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  1. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Improved uniquely decodable code pair bounds for unbalanced pairs. Unpublished. Google Scholar
  2. 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 für Informatik, 2015. URL:,
  3. Per Austrin, Mikko Koivisto, Petteri Kaski, and Jesper Nederlof. Dense subset sum may be the hardest. CoRR, abs/1508.06019, 2015. URL:
  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. 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
  7. Abraham Flaxman and Bartosz Przydatek. Solving medium-density subset sum problems in expected polynomial time. In Volker Diekert and Bruno Durand, editors, STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany, February 24-26, 2005, Proceedings, volume 3404 of Lecture Notes in Computer Science, pages 305-314. Springer, 2005. URL:,
  8. Open problems for FPT school 2014. URL:
  9. Ellis Horowitz and Sartaj Sahni. Computing partitions with applications to the knapsack problem. J. ACM, 21(2):277-292, 1974. URL:, URL:
  10. 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
  11. Russell Impagliazzo and Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology, 9(4):199-216, 1996. URL:,
  12. T. Kasami, Shu Lin, V.K. Wei, and Saburo Yamamura. Graph theoretic approaches to the code construction for the two-user multiple-access binary adder channel. IEEE Transactions on Information Theory, 29(1):114-130, 1983. URL:
  13. Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Homomorphic hashing for sparse coefficient extraction. In Dimitrios M. Thilikos and Gerhard J. Woeginger, editors, Parameterized and Exact Computation - 7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceedings, volume 7535 of Lecture Notes in Computer Science, pages 147-158. Springer, 2012. URL:,
  14. Jeffrey C. Lagarias and Andrew M. Odlyzko. Solving low-density subset sum problems. J. ACM, 32(1):229-246, 1985. Google Scholar
  15. Daniel Lokshtanov and Jesper Nederlof. Saving space by algebraization. In Leonard J. Schulman, editor, STOC, pages 321-330. ACM, 2010. Google Scholar
  16. 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. URL:,
  17. Or Ordentlich and Ofer Shayevitz. A VC-dimension-based outer bound on the zero-error capacity of the binary adder channel. CoRR, abs/1412.8670, 2014. URL:
  18. 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
  19. Christian Sleger and Alex Grant. Coordinated Multiuser Communications. Springer, 2006. Google Scholar
  20. Gerhard J. Woeginger. Open problems around exact algorithms. Discrete Appl. Math., 156(3):397-405, 2008. URL:,
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