License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.29
URN: urn:nbn:de:0030-drops-147223
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Brakerski, Zvika ; Stephens-Davidowitz, Noah ; Vaikuntanathan, Vinod

On the Hardness of Average-Case k-SUM

LIPIcs-APPROX29.pdf (0.7 MB)


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.

BibTeX - Entry

  author =	{Brakerski, Zvika and Stephens-Davidowitz, Noah and Vaikuntanathan, Vinod},
  title =	{{On the Hardness of Average-Case k-SUM}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{29:1--29:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-147223},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.29},
  annote =	{Keywords: k-SUM, fine-grained complexity, average-case hardness}

Keywords: k-SUM, fine-grained complexity, average-case hardness
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue Date: 2021
Date of publication: 15.09.2021

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