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Faster Approximation Schemes for the Two-Dimensional Knapsack Problem

Published: 08 August 2019 Publication History

Abstract

For geometric optimization problems we often understand the computational complexity on a rough scale, but not very well on a finer scale. One example is the two-dimensional knapsack problem for squares. There is a polynomial time (1+ε)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1/ε, i.e., Ω (n221/ε). A double or triple exponential dependence on 1/ε is inherent in how this and other algorithms for other geometric problems work. In this article, we present an efficient PTAS (EPTAS) for knapsack for squares, i.e., a (1+ε)-approximation algorithm with a running time of Oε(1)⋅ nO(1). In particular, the exponent of n in the running time does not depend on ε at all! Since there can be no fully polynomial time approximation scheme (FPTAS) for the problem (unless P = NP), this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω (221/ε) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS, each of these steps needs a running time of Ω (n221/ε) and we improve both to Oε(1)⋅ nO(1).
We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1+ε)-resource augmentation. We improve the previous double-exponential PTAS to an EPTAS and compute even a solution with optimal weight, while the previous PTAS computes only an approximation.

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Cited By

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  • (2024)On the Two-Dimensional Knapsack Problem for Convex PolygonsACM Transactions on Algorithms10.1145/364439020:2(1-38)Online publication date: 13-Apr-2024
  • (2022)Knapsack problems — An overview of recent advances. Part II: Multiple, multidimensional, and quadratic knapsack problemsComputers & Operations Research10.1016/j.cor.2021.105693143(105693)Online publication date: Jul-2022
  • (2021)Approximating Geometric Knapsack via L-packingsACM Transactions on Algorithms10.1145/347371317:4(1-67)Online publication date: 31-Oct-2021
  • Show More Cited By

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  1. Faster Approximation Schemes for the Two-Dimensional Knapsack Problem

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    cover image ACM Transactions on Algorithms
    ACM Transactions on Algorithms  Volume 15, Issue 4
    October 2019
    297 pages
    ISSN:1549-6325
    EISSN:1549-6333
    DOI:10.1145/3351875
    Issue’s Table of Contents
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    Publication History

    Published: 08 August 2019
    Accepted: 01 May 2019
    Revised: 01 December 2018
    Received: 01 September 2017
    Published in TALG Volume 15, Issue 4

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    Author Tags

    1. EPTAS
    2. Geometric knapsack problem
    3. approximation algorithm

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    Cited By

    View all
    • (2024)On the Two-Dimensional Knapsack Problem for Convex PolygonsACM Transactions on Algorithms10.1145/364439020:2(1-38)Online publication date: 13-Apr-2024
    • (2022)Knapsack problems — An overview of recent advances. Part II: Multiple, multidimensional, and quadratic knapsack problemsComputers & Operations Research10.1016/j.cor.2021.105693143(105693)Online publication date: Jul-2022
    • (2021)Approximating Geometric Knapsack via L-packingsACM Transactions on Algorithms10.1145/347371317:4(1-67)Online publication date: 31-Oct-2021
    • (2020)Framework for ER-Completeness of Two-Dimensional Packing Problems2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00098(1014-1021)Online publication date: Nov-2020

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