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arXiv · 2208.02664

Makespan Scheduling of Unit Jobs with Precedence Constraints in $O(1.995^n)$ time

Jesper Nederlof·Céline M. F. Swennenhuis·Karol Węgrzycki
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Abstract

In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$. We present an algorithm for this problem that runs in $O(1.995^n)$ time. Before our work, even for $m=3$ machines the best known algorithms ran in $O^\ast(2^n)$ time. In contrast, our algorithm works when the number of machines $m$ is unbounded. A crucial ingredient of our approach is an algorithm with a runtime that is only single-exponential in the vertex cover of the comparability graph of the precedence constraint graph. This heavily relies on insights from a classical result by Dolev and Warmuth (Journal of Algorithms 1984) for precedence graphs without long chains.

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Jesper Nederlof, Céline M. F. Swennenhuis, Karol Węgrzycki. 2022-08-04. Makespan Scheduling of Unit Jobs with Precedence Constraints in $O(1.995^n)$ time. https://arxiv.org/abs/2208.02664

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