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

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 \(\varepsilon > 0\).