Minimizing Makespan in Sublinear Time via Weighted Random Sampling

TL;DR

This paper introduces two sublinear time approximation algorithms for makespan minimization scheduling using weighted random sampling, achieving near-optimal solutions with sketch schedules.

Contribution

The paper presents novel sublinear time algorithms for scheduling that provide approximate solutions and sketch schedules, applicable when the number of jobs is known or unknown.

Findings

Both algorithms achieve a (1+3ε)-approximation to the optimal makespan.

The algorithms operate in sublinear time, depending on the number of jobs and machines.

An implementation for weighted random sampling with O(log n) uniform samples is provided.

Abstract

We consider the classical makespan minimization scheduling problem where $n$ jobs must be scheduled on $m$ identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where $n$ is known and the other for the case where $n$ is unknown. Both algorithms not only give a $(1+3\epsilon)$-approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where $n$ is known and draws samples in a single round under weighted random sampling, has a running time of $\tilde{O}(\tfrac{m^5}{\epsilon^4} \sqrt{n}+A(\ceiling{m\over \epsilon}, {\epsilon} ))$, where $A(\mathcal{N}, \alpha)$ is the time complexity of any $(1+\alpha)$-approximation scheme for the makespan minimization of $\mathcal{N}$ jobs. The second algorithm addresses the case where $n$ is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time $\tilde{O}\left( \tfrac{m^5} {\epsilon^4} \sqrt{n} + A(\ceiling{m\over \epsilon}, {\epsilon} )\right)$. We also provide an implementation that generates a weighted random sample using $O(\log n)$ uniform random samples.