1. A Unified Framework for Max -Min and Min-Max Fairness with application
Bozidar Radunoic, Member, IEEE, and
Jean-Yves Le Boudec, Fellow, IEEE
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 5, OCTOBER 2007
R 謝友仁
R 王猷順
R 陳冠瑋
指導老師：林永松 老師

2. A Unified Framework for Max -Min and Min-Max Fairness with application
Author
Bozidar Radunoic
Received the B.S. degree in electrical engineering from the University of Belgrade, Serbia, in 1999, and the Doctorate in 2005 from EPFL, Switzerland.
Participated in the Swiss NCCR project terminodes.org.
第一位作者在1999年時取得電機學士，到2005年時於EPFL取得博士學位，除此之外，該作者也參與Swiss NCCR project。他的研究領域在於architecture and performance of wireless ad-hoc networks
His interests are in the architecture and performance of wireless ad-hoc networks.
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A Unified Framework for Max -Min and Min-Max Fairness with application

3. A Unified Framework for Max -Min and Min-Max Fairness with application
Author
Jean-Yves Le Boudec
Graduated from Ecole Normale Superieure de Saint-Cloud, Paris, and received the Doctorate in 1984 from the University of Rennes, France.
In 1987, he joined Bell Northern Research, Ottawa, Canada.
In 1988, he joined the IBM Zurich Research Laboratory.
第二位作者在1984年時取得博士學位，1987年加入Bell Northern Research，1988年十，也參與IBM Zurich Research Laboratory，至1994年，於EPFL擔任教授的職務
In 1994, he became a Professor at EPFL, where he is now a full Professor.
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A Unified Framework for Max -Min and Min-Max Fairness with application

4. A Unified Framework for Max -Min and Min-Max Fairness with application
Abstract
Max-min fairness is widely used
Based on the notion of bottlenecks
A unifying treatment of max-min fairness
First, we observe that the existence of max-min fairness
A geometric property of the set of feasible allocations
點出Max-min主要是利用bottleneck的觀念，在尋找的過程中以bottleneck為跳板
該篇也對Max-min fairness 提出一個統一的架構
就Max-min fairness 的存在性而言，是以feasible allocations的幾何特性決定
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A Unified Framework for Max -Min and Min-Max Fairness with application

5. A Unified Framework for Max -Min and Min-Max Fairness with application
Abstract (cont’d)
Second, we give a general purpose centralized algorithm
Max-min Programming
Complexity is the order of N linear programming steps in RN
Free disposal property and Water Filling Algorithm
Mutatis mutandis to min-max fairness
介紹名詞Max-min Programming
詳細的複雜度分析在後面介紹
稍微題Free disposal 、Water Filling 和Max-min Programming的關係
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A Unified Framework for Max -Min and Min-Max Fairness with application

6. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

7. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

8. A Unified Framework for Max -Min and Min-Max Fairness with application
Max-Min Fairness
Allocating as much as possible to users with low rates x1 7
用圖代Water Filling的觀念 8 3 x2
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9. Microeconomic Approaches to Fairness
Max-min fairness is closely related to leximin ordering.
Free disposal property
A bottleneck argument can be made
稍為題leximin ordering並以前頁圖的斜線為例
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A Unified Framework for Max -Min and Min-Max Fairness with application

10. Bottleneck and Water-Filling
If each flow has a bottleneck link, then the rate allocation is max-min fair.
The bottleneck argument is often used to prove the existence of max-min fairness.
說明Bottleneck 和 Water-Filling的關係
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A Unified Framework for Max -Min and Min-Max Fairness with application

11. Water-filling algorithm (WF)
Rates of all flows are increased at the same pace, until one or more links are saturated.
正試舉例介紹Water-filling 1
0.5
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A Unified Framework for Max -Min and Min-Max Fairness with application

12. When Bottleneck and Water-Filling Become Less Obvious
Point-to-point multi-path routing scenario.
All links have capacity 1.
介紹Water-filling不能直接解決(須轉換)的例子
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A Unified Framework for Max -Min and Min-Max Fairness with application

13. When Bottleneck and Water-Filling Become Less Obvious (cont’d)
Re-interpret the original multi-path problem as a virtual single path problem.
x1, x2, y1, y2≥0, x1=y1+y2, y2+x2≤1, y1≤1
x2≤1, x1+x2≤2 y1 y2
描述轉換方法，並recall water-filling的斜線精神 x2
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A Unified Framework for Max -Min and Min-Max Fairness with application

14. When Bottleneck and Water-Filling Do Not Work
Equalize load on the servers while satisfying the capacity constraints.
Min-max fair
Cannot decrease a load on a server without increasing a load of another server that already has a higher load. 7 8 3
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A Unified Framework for Max -Min and Min-Max Fairness with application

15. A Unified Framework for Max -Min and Min-Max Fairness with application
Our Findings
The existence of max-min fairness
On algorithms to locate the max-min fair allocation
Max-min Programming (MP)
Relation between the general MP algorithm and the existing WF algorithm
Free disposal property
Form of feasible set 小結
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A Unified Framework for Max -Min and Min-Max Fairness with application

16. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

17. Definitions and Uniqueness
If is a min-max fair vector on , then is max-min fair on and vice versa
If a max-min fair vector exists on a set , then it is unique
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A Unified Framework for Max -Min and Min-Max Fairness with application

18. Max-Min Fairness and Leximin Ordering
Leximin maximum is not necessarily unique, if a vector is leximin maximal, it is also Pareto optimal
If a max-min fair vector exists on a set , then it is the unique leximin maximal vector on
it follows that the max-min fair vector, if it exists, is Pareto optimal
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A Unified Framework for Max -Min and Min-Max Fairness with application

19. Existence and Max-Min Achievable Sets
Max-min fair vector does not exist on all feasible sets
A set is max-min achievable if there exists a max-min fair vector on
In the example on the left, both points (1, 3) and (3, 1) are leximin maximal in this example.
In the example on the right, point (3, 1) is the single leximin maximal point.
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A Unified Framework for Max -Min and Min-Max Fairness with application

20. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

21. The Max-Min Programming (MP) Algorithm
finds the smallest component of the max-min fair vector
by maximizing the minimal coordinate
minimal coordinate is fixed, and the dimension corresponding to the minimal coordinate is removed
Repeated until all coordinates are fixed
vector obtained in such way is indeed the max-min fair one
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A Unified Framework for Max -Min and Min-Max Fairness with application

22. The Max-Min Programming (MP) Algorithm (Cont’d)
finds the smallest component
minimal coordinate is fixed
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A Unified Framework for Max -Min and Min-Max Fairness with application

23. The Max-Min Programming (MP) Algorithm (Cont’d)
maximizes the minimal coordinate in each step
until all coordinates are processed
If is compact and max-min achievable, the above algorithm terminates and finds the max-min fair vector on in at most steps.
第一部分
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A Unified Framework for Max -Min and Min-Max Fairness with application

24. Max-Min Programming Numerical Example 1
Step 1:
Step 2:
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A Unified Framework for Max -Min and Min-Max Fairness with application

25. Max-Min Programming Numerical Example 2
Step 1:
The min-max fair rate allocation is thus (4,3)
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A Unified Framework for Max -Min and Min-Max Fairness with application

26. Max-Min Programming with Max-min does not exist
max-min fair allocation does not exist
Step 1:
This point is neither leximin maximal, nor Pareto optimal
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A Unified Framework for Max -Min and Min-Max Fairness with application

27. The Water-Filling (WF) Algorithm
free disposal applies to sets where each coordinate is independently lower-bounded
Let be a max-min achievable set that satisfies the free disposal property. Then, at every step, the solutions to problems and are the same
WF terminates and returns the same result as MP, namely the max-min fair vector if it exists
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A Unified Framework for Max -Min and Min-Max Fairness with application

28. The Water-Filling (WF) Algorithm (Cont’d)
set each xi to be the smallest
component or its original minimum
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A Unified Framework for Max -Min and Min-Max Fairness with application

29. The Water-Filling (WF) Algorithm (Cont’d)
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A Unified Framework for Max -Min and Min-Max Fairness with application

30. Water-Filling Algorithm Numerical Example
Free disposal property is a sufficient but not a necessary condition for MP to degenerate to WF
C1 = 3, C2 = 3, C3 = 4, X1 + X2 ≥ 3
same shape and orientation as in Fig. 4, but it is translated to the left such that it touches both X2 and X1 axes
Without the free disposal property, still solved by WF in the single step
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A Unified Framework for Max -Min and Min-Max Fairness with application

31. Complexity of the Algorithms in Case of Linear Constraints
Χ is a n-dimensional feasible set defined by m linear inequalities
Max-Min Programming Complexity
the overall complexity is O(nLP(n,m))
LP(n,m) is the complexity of linear programming
Linear programming
the worst case: exponential complexity
most practical cases: polynomial complexity
Water-Filling Algorithm Complexity
N steps, each of complexity O(m), overall O(nm)
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A Unified Framework for Max -Min and Min-Max Fairness with application

32. Complexity of the Algorithms in Case of Linear Constraints
Χ is defined implicitly, with an ι–dimensional slack variable (ex: multi-path case)
Max-Min Programming on implicit sets
The complexity is O(nLP(n,m))
Water-Filling Algorithm on implicit sets
it might be possible to construct an implicitly defined feasible set that cannot be converted to an explicit form in a polynomial time
paper interested in explicitly finding the values of the slack variables at the max-min fair vector, and the complexity is O(nm)
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A Unified Framework for Max -Min and Min-Max Fairness with application

33. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

34. A Unified Framework for Max -Min and Min-Max Fairness with application
Example Scenarios
Load Distribution in P2P Systems
a minimal rate a user needs to achieve
an upper bound on each flow given by a network topology and link capacities
Maximum Lifetime Sensor Networks
to minimize the average transmitting powers of sensors
we look for min-max fair vector of average power consumptions of sensors.
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A Unified Framework for Max -Min and Min-Max Fairness with application

35. Load Distribution in P2P Systems
represent the feasible rate set as
be the total loads on the servers
the flows from the servers to clients
the total traffic received by clients
the capacities of links
the minimum required rates of the flows
where A, B, C ≥ 0 are arbitrary matrices defined by network topology and routing.
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A Unified Framework for Max -Min and Min-Max Fairness with application

36. Load Distribution in P2P Systems
Each server is interested in minimizing its own load
It is natural to look for the min-max fair vector
Since set is convex, it is min-max achievable
Use Max-min Programming to solve this problem Y1 Y2
A代表為所有capacity的矩陣，B代表所有server的loading矩陣，C代表為client所收到的總traffic矩陣。
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A Unified Framework for Max -Min and Min-Max Fairness with application

37. Maximum Lifetime Sensor Networks
A network has a certain minimal amount of data to convey to a sink
consider different scheduling and routing strategies that achieve this goal
Each of these strategies yields different average power consumptions
Look for min-max fair vector of average power consumptions of sensors
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A Unified Framework for Max -Min and Min-Max Fairness with application

38. Maximum Lifetime Sensor Networks
The maximum rate of information can achieve is
Denote with the average rate of sd link throughout a schedule
Χ is a set of feasible , that is such that there exists a schedule and power allocations that achieve those rates
calculate the average power dissipated by a node during a schedule
Time
Power
Rate
Total interference
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A Unified Framework for Max -Min and Min-Max Fairness with application

39. Maximum Lifetime Sensor Networks
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A Unified Framework for Max -Min and Min-Max Fairness with application

40. Maximum Lifetime Sensor Networks
P = 1
N = 1
hS1D1 = 1
hS1D2 = 1
hS2D1 = 10
hS2D2 = 0.7
M1 = 0.6
M2 = 0.4
P1=0, P2= -0.62; min(P1, P2)=-0.62
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A Unified Framework for Max -Min and Min-Max Fairness with application

41. Agenda Introduction Max-min and Min-max Fairness
Max-min Programming and Water-filling
Example Scenarios
Conclusion

42. A Unified Framework for Max -Min and Min-Max Fairness with application
Conclusion
We have elucidated the role of bottleneck arguments in the water-filling algorithm, and explained the relation to the free disposal property
We have given a general purpose algorithm (MP) for computing the max-min fair vector whenever it exists, and showed that it degenerates to the classical water-filling algorithm, when free disposal property holds
We have focused on centralized algorithms for calculating max-min and min-max fair allocations
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A Unified Framework for Max -Min and Min-Max Fairness with application

43. Thanks for your listening