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On the Hardness of Average-Case k-SUM

Authors Zvika Brakerski, Noah Stephens-Davidowitz, Vinod Vaikuntanathan

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Author Details

Zvika Brakerski
  • Weizmann Institute of Science, Rehovot, Israel
Noah Stephens-Davidowitz
  • Cornell University, Ithaca, NY, USA
Vinod Vaikuntanathan
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Brakerski, Zvika: Supported by the Binational Science Foundation (Grant No. 2016726), and by the European Union Horizon 2020 Research and Innovation Program via ERC Project REACT (Grant 756482) and via Project PROMETHEUS (Grant 780701).
  • Stephens-Davidowitz, Noah: Supported in part by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, Microsoft Faculty Fellowship and an MIT/IBM grant.
  • Vaikuntanathan, Vinod: Supported in part by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, Microsoft Faculty Fellowship and an MIT/IBM grant.

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Zvika Brakerski, Noah Stephens-Davidowitz, and Vinod Vaikuntanathan. On the Hardness of Average-Case k-SUM. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.29

Abstract

In this work, we show the first worst-case to average-case reduction for the classical k-SUM problem. A k-SUM instance is a collection of m integers, and the goal of the k-SUM problem is to find a subset of k integers that sums to 0. In the average-case version, the m elements are chosen uniformly at random from some interval [-u,u].
We consider the total setting where m is sufficiently large (with respect to u and k), so that we are guaranteed (with high probability) that solutions must exist. In particular, m = u^{Ω(1/k)} suffices for totality. Much of the appeal of k-SUM, in particular connections to problems in computational geometry, extends to the total setting.
The best known algorithm in the average-case total setting is due to Wagner (following the approach of Blum-Kalai-Wasserman), and achieves a running time of u^{Θ(1/log k)} when m = u^{Θ(1/log k)}. This beats the known (conditional) lower bounds for worst-case k-SUM, raising the natural question of whether it can be improved even further. However, in this work, we show a matching average-case lower bound, by showing a reduction from worst-case lattice problems, thus introducing a new family of techniques into the field of fine-grained complexity. In particular, we show that any algorithm solving average-case k-SUM on m elements in time u^{o(1/log k)} will give a super-polynomial improvement in the complexity of algorithms for lattice problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • k-SUM
  • fine-grained complexity
  • average-case hardness

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