No Polynomial Kernels for Knapsack
Klaus Heeger, Danny Hermelin, Matthias Mnich, Dvir Shabtay

TL;DR
This paper proves that the Knapsack problem does not have polynomial kernels for certain parameters, using advanced lower bound techniques, but admits a polynomial kernel when combining those parameters.
Contribution
It establishes non-existence of polynomial kernels for Knapsack with specific parameters and introduces novel techniques for kernelization lower bounds.
Findings
No polynomial kernel for weight parameter $w_{ ext{#}}$
No polynomial kernel for profit parameter $p_{ ext{#}}$
Polynomial kernel exists for combined parameter $w_{ ext{#}}+p_{ ext{#}}$
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 reduces any instance of Knapsack to an equivalent instance of size at most in polynomial time, for some computable function . When then we call such a reduction a polynomial kernel. Our study focuses on two natural parameters for Knapsack: The number of different item weights , and the number of different item profits . Our main technical contribution is a proof showing that Knapsack…
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