# Linear Time Algorithms for Multiple Cluster Scheduling and Multiple Strip Packing

## TL;DR

This paper introduces an optimal linear-time approximation algorithm for multiple cluster scheduling and strip packing problems, improving computational efficiency while maintaining a 2-approximation ratio, and explores practical heuristics with better ratios in specific cases.

## Contribution

The authors develop an $O(n)$ time algorithm achieving a 2-approximation for both problems, improving over previous algorithms with much higher running times, and propose practical heuristics with improved ratios for certain instances.

## Key findings

- An $O(n)$ algorithm with a 2-approximation ratio for both problems.
- A practical $O(n \log n)$ algorithm with a 9/4-approximation for specific instances.
- The approach of scheduling on one cluster then distributing can be applied broadly.

## Abstract

We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than $2$ unless $P = NP$. In this paper, we present an algorithm with approximation ratio $2$ and running time $O(n)$ for both problems. While a $2$ approximation was known before, the running time of the algorithm is at least $\Omega(n^{256})$ in the worst case. Therefore, an $O(n)$ algorithm is surprising and the best possible. We archive this result by calling an AEPTAS with approximation guarantee $(1+\varepsilon)OPT +p_{\max}$ and running time of the form $O(n\log(1/\varepsilon)+ f(1/\varepsilon))$ with a constant $\varepsilon$ to schedule the jobs on a single cluster. This schedule is then distributed on the $N$ clusters in $O(n)$. Moreover, this distribution technique can be applied to any variant of of Multi Cluster Scheduling for which there exists an AEPTAS with additive term $p_{\max}$. While the above result is strong from a theoretical point of view, it might not be very practical due to a large hidden constant caused by calling an AEPTAS with a constant $\varepsilon \geq 1/8$ as subroutine. Nevertheless, we point out that the general approach of finding first a schedule on one cluster and then distributing it onto the other clusters might come in handy in practical approaches. We demonstrate this by presenting a practical algorithm with running time $O(n\log(n))$, with out hidden constants, that is a $9/4$-approximation for one third of all possible instances, i.e, all instances where the number of clusters is dividable by $3$, and has an approximation ratio of at most $2.3$ for all instances with at least $9$ clusters.