ebook img

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

62 Pages·2010·0.49 MB·English
by  
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

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

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

Description:
The uniformly related machine scheduling model with sum of completion .. The Nash equilibrium concept assumes complete selfishness and no
See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.