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

TL;DR

This paper introduces a faster exact algorithm for scheduling unit jobs with precedence constraints on multiple machines, achieving a runtime of O(1.995^n), improving over previous exponential algorithms.

Contribution

The paper presents a novel algorithm with sub-2 exponential runtime for a classical NP-hard scheduling problem with unbounded machines.

Findings

Achieves O(1.995^n) runtime for the problem

Extends algorithms to unbounded machine scenarios

Uses vertex cover approach for precedence graph analysis

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. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-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 of machines $m$ is unbounded. A crucial ingredient of our approach is an algorithm with a runtime that is only single-exponential in the vertex cover of the comparability graph of the precedence constraint graph. This heavily relies on insights from a classical result by Dolev and Warmuth (Journal of Algorithms 1984) for precedence graphs without long chains.