Polynomial Kernels for Weighted Problems

TL;DR

This paper demonstrates the existence of polynomial kernels for weighted NP-hard problems like Subset Sum and Knapsack, using number compression techniques, and extends these results to other related problems and parameterizations.

Contribution

It provides the first deterministic polynomial kernelizations for weighted Subset Sum and Knapsack parameterized by item count, and extends kernelization techniques to polynomial integer programs.

Findings

Polynomial kernels for weighted Subset Sum and Knapsack with respect to item count.

Polynomial kernels for weighted problems using number compression techniques.

Kernelization results for polynomial integer programs.

Abstract

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs.