No Polynomial Kernels for Knapsack

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 $k$ reduces any instance of Knapsack to an equivalent instance of size at most $f(k)$ in polynomial time, for some computable function $f(\cdot)$. 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 of different item weights $w_{\#}$, and the number of different item profits $p_{\#}$. 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 the Knapsack admits a polynomial kernel for the combined parameter $w_{\#}+p_{\#}$.