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

TL;DR

This paper introduces a subexponential time algorithm for the classical makespan scheduling problem with unit jobs and precedence constraints, significantly improving the computational complexity for certain machine regimes.

Contribution

It presents the first subexponential algorithm for the problem, resolving a long-standing open question and providing faster solutions for specific machine counts.

Findings

Algorithm runs in $(1+rac{n}{m})^{O(\sqrt{nm})}$ time for certain regimes.

For $m=Θ(n)$, the algorithm runs in $O(1.997^n)$ time.

No previous algorithms achieved $O((2-\varepsilon)^n)$ time for this problem.

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 $m=\Theta(n)$ 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 $\varepsilon > 0$.