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.ESA.2023.24
URN: urn:nbn:de:0030-drops-186776
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Bringmann, Karl ; Cassis, Alejandro

Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution

LIPIcs-ESA-2023-24.pdf (1 MB)


We revisit the classic 0-1-Knapsack problem, in which we are given n items with their weights and profits as well as a weight budget W, and the goal is to find a subset of items of total weight at most W that maximizes the total profit. We study pseudopolynomial-time algorithms parameterized by the largest profit of any item p_{max}, and the largest weight of any item w_max. Our main result are algorithms for 0-1-Knapsack running in time Õ(n w_max p_max^{2/3}) and Õ(n p_max w_max^{2/3}), improving upon an algorithm in time O(n p_max w_max) by Pisinger [J. Algorithms '99]. In the regime p_max ≈ w_max ≈ n (and W ≈ OPT ≈ n²) our algorithms are the first to break the cubic barrier n³.
To obtain our result, we give an efficient algorithm to compute the min-plus convolution of near-convex functions. More precisely, we say that a function f : [n] ↦ ℤ is Δ-near convex with Δ ≥ 1, if there is a convex function f ̆ such that f ̆(i) ≤ f(i) ≤ f ̆(i) + Δ for every i. We design an algorithm computing the min-plus convolution of two Δ-near convex functions in time Õ(nΔ). This tool can replace the usage of the prediction technique of Bateni, Hajiaghayi, Seddighin and Stein [STOC '18] in all applications we are aware of, and we believe it has wider applicability.

BibTeX - Entry

  author =	{Bringmann, Karl and Cassis, Alejandro},
  title =	{{Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-186776},
  doi =		{10.4230/LIPIcs.ESA.2023.24},
  annote =	{Keywords: Knapsack, Fine-Grained Complexity, Min-Plus Convolution}

Keywords: Knapsack, Fine-Grained Complexity, Min-Plus Convolution
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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