Weighted completion time minimization for capacitated parallel machines

TL;DR

This paper introduces the first polynomial-time constant approximation algorithm for minimizing weighted completion times on capacitated parallel machines, addressing a complex NP-hard scheduling problem with practical relevance in cloud computing.

Contribution

It proposes novel heuristic algorithms with provable approximation guarantees for weighted completion time minimization on capacitated parallel machines, including a hybrid approach for multiple machines.

Findings

Bounded approximation ratio based on resource demand ratios

Developed a hybrid algorithm combining two scheduling strategies

First polynomial-time constant approximation for this problem

Abstract

We consider the weighted completion time minimization problem for capacitated parallel machines, which is a fundamental problem in modern cloud computing environments. We study settings in which the processed jobs may have varying duration, resource requirements and importance (weight). Each server (machine) can process multiple concurrent jobs up to its capacity. Due to the problem's $\mathcal{NP}$-hardness, we study heuristic approaches with provable approximation guarantees. We first analyze an algorithm that prioritizes the jobs with the smallest volume-by-weight ratio. We bound its approximation ratio with a decreasing function of the ratio between the highest resource demand of any job to the server's capacity. Then, we use the algorithm for scheduling jobs with resource demands equal to or smaller than 0.5 of the server's capacity in conjunction with the classic weighted shortest processing time algorithm for jobs with resource demands higher than 0.5. We thus create a hybrid, constant approximation algorithm for two or more machines. We also develop a constant approximation algorithm for the case with a single machine. This research is the first, to the best of our knowledge, to propose a polynomial-time algorithm with a constant approximation ratio for minimizing the weighted sum of job completion times for capacitated parallel machines.