On Strong NP-Completeness of Rational Problems

TL;DR

This paper proves that many classical combinatorial optimization problems become strongly NP-complete when weights and profits are rational numbers, ruling out pseudo-polynomial algorithms but allowing approximation schemes.

Contribution

It demonstrates that the complexity status of these problems changes significantly when rational data is considered, establishing their strong NP-completeness.

Findings

All studied problems are strongly NP-complete with rational data.

Pseudo-polynomial algorithms are unlikely to exist for these problems.

Fully polynomial-time approximation schemes are still possible.

Abstract

The computational complexity of the partition, 0-1 subset sum, unbounded subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple variants were studied in numerous papers in the past where all the weights and profits were assumed to be integers. We re-examine here the computational complexity of all these problems in the setting where the weights and profits are allowed to be any rational numbers. We show that all of these problems in this setting become strongly NP-complete and, as a result, no pseudo-polynomial algorithm can exist for solving them unless P=NP. Despite this result we show that they all still admit a fully polynomial-time approximation scheme.