# Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

## TL;DR

This paper establishes that the parallel task scheduling problem with four machines is strongly NP-complete and improves the inapproximability bounds for strip packing, narrowing the gap between known approximation algorithms and hardness results.

## Contribution

It proves the NP-completeness for four machines in parallel task scheduling and enhances the inapproximability bound for strip packing from 12/11 to 5/4.

## Key findings

- NP-completeness for m=4 in parallel task scheduling
- Improved inapproximability bound for strip packing to 5/4
- Reduction from 3-Partition problem used in proofs

## Abstract

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition.