Abstract
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 NPhardness. When release dates are identical there is also a gap: the problem remains strongly NPhard and the best known approximation algorithm has a ratio of e+\epsilon (running in quasipolynomial time). We reduce the latter gap by giving a QPTAS if the numbers in the input are quasipolynomially bounded, ruling out the existence of an APXhardness proof unless NP\subseteq DTIME(2^polylog(n)). Our techniques are based on the QPTAS known for the UFPCover 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 UFPCover and use a sophisticated divideandconquer procedure with interdependent nonsymmetric subproblems.
We also present a pseudopolynomial time approximation scheme for two variants of UFPCover. 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.
BibTeX  Entry
@InProceedings{antoniadis_et_al:LIPIcs:2017:7457,
author = {Antonios Antoniadis and Ruben Hoeksma and Julie Mei{\ss}ner and Jos{\'e} Verschae and Andreas Wiese},
title = {{A QPTAS for the General Scheduling Problem with Identical Release Dates}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {31:131:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7457},
URN = {urn:nbn:de:0030drops74575},
doi = {10.4230/LIPIcs.ICALP.2017.31},
annote = {Keywords: Generalized Scheduling, QPTAS, Unsplittable Flows}
}
Keywords: 

Generalized Scheduling, QPTAS, Unsplittable Flows 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 