Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Authors Jonathan Allcock , Yassine Hamoudi , Antoine Joux, Felix Klingelhöfer, Miklos Santha

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Author Details

Jonathan Allcock
  • Tencent Quantum Laboratory, Hong Kong, China
Yassine Hamoudi
  • Simons Institute for the Theory of Computing, University of California, Berkeley, CA, USA
Antoine Joux
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Felix Klingelhöfer
  • G-SCOP, Université Grenoble Alpes, France
Miklos Santha
  • Centre for Quantum Technologies and MajuLab, National University of Singapore, Singapore


JA thanks Shengyu Zhang for helpful discussions during the course of this work.

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Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Quantum algorithm
  • classical algorithm
  • dynamic programming
  • representation technique
  • subset-sum
  • equal-sum
  • shifted-sum


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