Price of Anarchy for Machine Scheduling Games with Sum of Completion Times Objective

62 Pages·2010·0.49 MB·English

Master Thesis
Price of Anarchy for Machine
Scheduling Games with Sum of
Completion Times Objective
Ruben Hoeksma
Graduation Committee:
prof.dr. M.J. Uetz
prof.dr. J.L. Hurink
dr. B. Manthey
October 26, 2010

Voor Nan

Preface
This master thesis is the final result of my research on the price of anarchy for
uniformly related machine scheduling with sum of completion times objective.
The research was done at the chair of Discrete Mathematics and Mathematical
Programming at the University of Twente. I would like to use this opportunity
to thank the people who supported me during the research for and the writing
of this thesis.
First of all I want to thank Marc Uetz for introducing me to the realm of algo-
rithmicgametheoryand,especially,forcomingupwiththeresearchtopicofthis
thesis. Alsomanythanksfortheconstructivecriticismandusefuldeliberations.
I would also like to thank the rest of the staff and the master and Ph.D. stu-
dents from the chair of Discrete Mathematics and Mathematical Programming
for a good working environment and interesting discussions when we were not
working.
Finally, I am very grateful to my family and my friends who supported me
during my whole study and, especially, while I was working on this thesis.
Ruben Hoeksma
Enschede, October 2010
v

Abstract
The uniformly related machine scheduling model with sum of completion times
objective is well known and its optimal solution is easy to find. However, this
solutionrequiresacentraladministratorthatschedulesthejobsonthemachines.
We look at this model from a game theoretical point of view and introduce for
each job a selfish agent, which chooses the machine which processes the job,
and is only interested in minimizing the completion time of its own job. Our
interest lies in the price of anarchy for this game. That is, the ratio between
the objective value of the optimal solution and the objective value of the Nash
equilibrium of the game.
Recent work hasshown thatfor themore generalunrelated machinescheduling
model, with the same objective, the price of anarchy is at least 3 and at most
4. For the uniformly related machine scheduling game, with only two machines
√
weproveanupperboundonthepriceofanarchyof 1+1 5≈1.6180. Wealso
2 2 √
construct an instance which has price of anarchy equal to 3 · 2+√3 ≈ 1.1830.
2 3+ 3
For the general case with any number of machines we show an upper bound of
3. Furthermore, we construct a set of instances, which price of anarchy can be
made arbitrarily close to e ≈1.5820.
e−1

Contents
1 Introduction 1
1.1 Machine scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Scheduling games . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The problem setting 7
2.1 Uniformly related machine scheduling . . . . . . . . . . . . . . . 7
2.2 Single machine model . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Optimal solution algorithm . . . . . . . . . . . . . . . . . . . . . 9
2.4 The scheduling game . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Equilibrium solution algorithm . . . . . . . . . . . . . . . . . . . 12
2.6 Robust price of anarchy . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Instances with large price of anarchy 17
3.1 Reducing the set of instances . . . . . . . . . . . . . . . . . . . . 17
3.2 Simple instances . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Instances with all jobs on the fastest machine in equilibrium . . . 23
4 Upper bounds on the price of anarchy 29
4.1 Known bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Unit speeds and unit lengths . . . . . . . . . . . . . . . . . . . . 29
(cid:80)
4.3 Upper bounds on the price of anarchy for the Q|| C game . . 33
j
5 Concluding remarks 43
A Used notation 45
ix
x Contents
B Proofs of analysis steps 47
C Q||(cid:80)C is (2,1)-smooth 49
j 2
Bibliography 52