Strong LP formulations for scheduling splittable jobs on unrelated machines

• 成果类型: Article
• 署名作者: Correa, Jose; Marchetti-Spaccamela, Alberto; Matuschke, Jannik; Stougie, Leen; Svensson, Ola; Verdugo, Victor; Verschae, Jose
• 署名单位: Universidad de Chile; Sapienza University Rome; Technical University of Berlin; Vrije Universiteit Amsterdam; Centrum Wiskunde & Informatica (CWI); Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Pontificia Universidad Catolica de Chile; Pontificia Universidad Catolica de Chile
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0831-8
• 发表日期: 2015
• 页码: 305-328
• 关键词: approximation algorithms; setup times

摘要

A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to , where is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than , which is larger than the inapproximability bound of 1.5 for the original problem.