Abstract
Makespan minimization in restricted assignment (Rp_{ij} in {p_j, infinity}C_{max}) is a classical problem in the field of machine scheduling. In a landmark paper, [Lenstra, Shmoys, and Tardos, Math. Progr. 1990] gave a 2approximation algorithm and proved that the problem cannot be approximated within 1.5 unless P=NP. The upper and lower bounds of the problem have been essentially unimproved in the intervening 25 years, despite several remarkable successful attempts in some special cases of the problem recently.
In this paper, we consider a special case called graphbalancing with light hyper edges, where heavy jobs can be assigned to at most two machines while light jobs can be assigned to any number of machines. For this case, we present algorithms with approximation ratios strictly better than 2. Specifically,
 Two job sizes: Suppose that light jobs have weight w and heavy jobs have weight W, and w < W. We give a 1.5approximation algorithm (note that the current 1.5 lower bound is established in an even more restrictive setting). Indeed, depending on the specific values of w and W, sometimes our algorithm guarantees sub1.5 approximation ratios.
 Arbitrary job sizes: Suppose that W is the largest given weight, heavy jobs have weights in the range of (beta W, W], where 4/7 <= beta < 1, and light jobs have weights in the range of (0,beta W]. We present a (5/3+beta/3)approximation algorithm.
Our algorithms are purely combinatorial, without the need of solving a linear program as required in most other known approaches.
BibTeX  Entry
@InProceedings{huang_et_al:LIPIcs:2016:6391,
author = {ChienChung Huang and Sebastian Ott},
title = {{A Combinatorial Approximation Algorithm for Graph Balancing with Light Hyper Edges}},
booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)},
pages = {49:149:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770156},
ISSN = {18688969},
year = {2016},
volume = {57},
editor = {Piotr Sankowski and Christos Zaroliagis},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6391},
URN = {urn:nbn:de:0030drops63919},
doi = {10.4230/LIPIcs.ESA.2016.49},
annote = {Keywords: Approximation Algorithms, Machine Scheduling, Graph Balancing, Combinatorial Algorithms}
}
Keywords: 

Approximation Algorithms, Machine Scheduling, Graph Balancing, Combinatorial Algorithms 
Collection: 

24th Annual European Symposium on Algorithms (ESA 2016) 
Issue Date: 

2016 
Date of publication: 

18.08.2016 