文章基本信息

标题：A QPTAS for the General Scheduling Problem with Identical Release Dates
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作者：Antonios Antoniadis ; Ruben Hoeksma ; Julie Mei{\ss}ner 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：80
页码：31:1-31:14
DOI：10.4230/LIPIcs.ICALP.2017.31
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The General Scheduling Problem (GSP) generalizes scheduling problems with sum of cost objectives such as weighted flow time and weighted tardiness. Given a set of jobs with processing times, release dates, and job dependent cost functions, we seek to find a minimum cost preemptive schedule on a single machine. The best known algorithm for this problem and also for weighted flow time/tardiness is an O(loglog P)-approximation (where P denotes the range of the job processing times), while the best lower bound shows only strong NP-hardness. When release dates are identical there is also a gap: the problem remains strongly NP-hard and the best known approximation algorithm has a ratio of e+\epsilon (running in quasi-polynomial time). We reduce the latter gap by giving a QPTAS if the numbers in the input are quasi-polynomially bounded, ruling out the existence of an APX-hardness proof unless NP\subseteq DTIME(2^polylog(n)). Our techniques are based on the QPTAS known for the UFP-Cover problem, a particular case of GSP where we must pick a subset of intervals (jobs) on the real line with associated heights and costs. If an interval is selected, its height will help cover a given demand on any point contained within the interval. We reduce our problem to a generalization of UFP-Cover and use a sophisticated divide-and-conquer procedure with interdependent non-symmetric subproblems. We also present a pseudo-polynomial time approximation scheme for two variants of UFP-Cover. For the case of agreeable intervals we give an algorithm based on a new dynamic programming approach which might be useful for other problems of this type. The second one is a resource augmentation setting where we are allowed to slightly enlarge each interval.
关键词：Generalized Scheduling; QPTAS; Unsplittable Flows