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arXiv · 2312.03495

A Subexponential Time Algorithm for Makespan Scheduling of Unit Jobs with Precedence Constraints

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. Using the standard 3-field notation, it is known as $Pm|\text{prec}, p_j=1|C_{\max}$. Settling the complexity of $Pm|\text{prec}, p_j=1|C_{\max}$ even for $m=3$ machines is the last open problem from the book of Garey and Johnson [GJ79] for which both upper and lower bounds on the worst-case running times of exact algorithms solving them remain essentially unchanged since the publication of [GJ79]. We present an algorithm for this problem that runs in $(1+\frac{n}{m})^{\mathcal{O}(\sqrt{nm})}$ time. This algorithm is subexponential when $m = o(n)$. In the regime of $m=Θ(n)$ we show an algorithm that runs in$\mathcal{O}(1.997^n)$ time. Before our work, even for $m=3$ machines there were no algorithms known that run in $\mathcal{O}((2-\varepsilon)^n)$ time for some $\varepsilon > 0$.

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Jesper Nederlof, Céline M. F. Swennenhuis, Karol Węgrzycki. 2023-12-06. A Subexponential Time Algorithm for Makespan Scheduling of Unit Jobs with Precedence Constraints. https://arxiv.org/abs/2312.03495

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