Fast Makespan Minimization via Short ILPs

TL;DR

This work shows that short ILP solvers can be harnessed to obtain fast pseudo-polynomial algorithms for makespan minimization on a fixed number of parallel machines and several related variants. By converting the standard ILP into a compact short-ILP form, notably for $Qm\mid\mid C_{\max}$, the authors achieve a runtime of $\widetilde{O}(p_{\max}^{2m+2}+n)$ and extend the approach to capacity, rejection, and release-date variants, as well as certain unrelated-machine and two-machine cases. The results improve upon prior pseudopolynomial bounds when $p_{\max}$ is sublinear in $n$, and they illustrate the broad applicability of short ILP techniques to scheduling problems. The work highlights the potential for furtherSpeedups by exploiting sparsity and applying short ILPs to additional scheduling models.

Abstract

Short integer linear programs are programs with a relatively small number of constraints. We show how recent improvements on the running-times of solvers for such programs can be used to obtain fast pseudo-polynomial time algorithms for makespan minimization on a fixed number of parallel machines, and other related variants. The running times of our algorithms are all of the form $\widetilde{O}(p^{O(1)}_{\max}+n)$ or $\widetilde{O}(p^{O(1)}_{\max} \cdot n)$, where $p_{\max}$ is the maximum processing time in the input. These improve upon the time complexity of previously known algorithms for moderate values of $p_{\max}$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2: EisenbrandWeismantel2020
• Theorem 3: JansenRohwedder
• Lemma 1
• proof
• Theorem 4
• Lemma 2
• proof
• Theorem 5
• Lemma 3