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In a classical scheduling problem, we are given a set of $n$ jobs of unitlength along with precedence constraints, and the goal is to find a schedule ofthese jobs on $m$ identical machines that minimizes the makespan. Using thestandard 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] forwhich both upper and lower bounds on the worst-case running times of exactalgorithms 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 issubexponential when $m = o(n)$. In the regime of $m=\Theta(n)$ we show analgorithm 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 $\varepsilon > 0$.

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