No Polynomial Kernels for Knapsack

Authors Klaus Heeger , Danny Hermelin , Matthias Mnich , Dvir Shabtay



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Klaus Heeger
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Danny Hermelin
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Matthias Mnich
  • Institute for Algorithms and Complexity, Hamburg University of Technology, Hamburg, Germany
Dvir Shabtay
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

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Klaus Heeger, Danny Hermelin, Matthias Mnich, and Dvir Shabtay. No Polynomial Kernels for Knapsack. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 83:1-83:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.83

Abstract

This paper focuses on kernelization algorithms for the fundamental Knapsack problem. A kernelization algorithm (or kernel) is a polynomial-time reduction from a problem onto itself, where the output size is bounded by a function of some problem-specific parameter. Such algorithms provide a theoretical model for data reduction and preprocessing and are central in the area of parameterized complexity. In this way, a kernel for Knapsack for some parameter k reduces any instance of Knapsack to an equivalent instance of size at most f(k) in polynomial time, for some computable function f. When f(k) = k^{O(1)} then we call such a reduction a polynomial kernel. Our study focuses on two natural parameters for Knapsack: The number w_# of different item weights, and the number p_# of different item profits. Our main technical contribution is a proof showing that Knapsack does not admit a polynomial kernel for any of these two parameters under standard complexity-theoretic assumptions. Our proof discovers an elaborate application of the standard kernelization lower bound framework, and develops along the way novel ideas that should be useful for other problems as well. We complement our lower bounds by showing that Knapsack admits a polynomial kernel for the combined parameter w_# ⋅ p_#.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Knapsack
  • polynomial kernels
  • compositions
  • number of different weights
  • number of different profits

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