Knapsack Problems: A Parameterized Point of View

TL;DR

This paper investigates the fixed-parameter tractability of various knapsack problems, analyzing how problem complexity can be managed by parameters like item count, profits, sizes, dimensions, and knapsacks, and exploring kernelization possibilities.

Contribution

It provides a comprehensive parameterized complexity analysis of knapsack variants, including kernelization and connections to linear programming and approximation algorithms.

Findings

Identifies parameters that influence tractability

Studies kernelization and size reduction possibilities

Connects parameterized problems to LP and approximation methods

Abstract

The knapsack problem (KP) is a very famous NP-hard problem in combinatorial optimization. Also its generalization to multiple dimensions named d-dimensional knapsack problem (d-KP) and to multiple knapsacks named multiple knapsack problem (MKP) are well known problems. Since KP, d-KP, and MKP are integer-valued problems defined on inputs of various informations, we study the fixed-parameter tractability of these problems. The idea behind fixed-parameter tractability is to split the complexity into two parts - one part that depends purely on the size of the input, and one part that depends on some parameter of the problem that tends to be small in practice. Further we consider the closely related question, whether the sizes and the values can be reduced, such that their bit-length is bounded polynomially or even constantly in a given parameter, i.e. the existence of kernelizations is studied. We discuss the following parameters: the number of items, the threshold value for the profit, the sizes, the profits, the number d of dimensions, and the number m of knapsacks. We also consider the connection of parameterized knapsack problems to linear programming, approximation, and pseudo-polynomial algorithms.