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

Abstract

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. This problemis well-known to be NP-hard for an unbounded number of machines. Using standard3-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 ofmachines $m$ is unbounded. A crucial ingredient of our approach is an algorithmwith a runtime that is only single-exponential in the vertex cover of thecomparability graph of the precedence constraint graph. This heavily relies oninsights from a classical result by Dolev and Warmuth (Journal of Algorithms1984) for precedence graphs without long chains.<br