1. Xinbing Wang*, Qian Zhang**
Combinatorial Auction with Time-Frequency Flexibility in Cognitive Radio Networks
Mo Dong*, Gaofei Sun*,
Xinbing Wang*, Qian Zhang**
*Department of Electronic Engineering
Shanghai Jiao Tong University, China
**Department of Comp. Sci. and Engin
HK Univ. of Sci. and Tech., HongKong
11/29/2018

2. Outline Introduction System Model and Problem Formulation
Solution of CAP Under General Flexibility Model
Solution of CAP Under Modified Model
Simulation Results and Future Works 2

3. Outline Introduction Background Objectives
System Model and Problem Formulation
Solution of CAP Under General Flexibility Model
Solution of CAP Under Modified Model
Simulation Results and Future Works 3

4. Background Spectrum Opportunity Allocation Mechanism SUs’ bids Payment
Dynamic Spectrum Access Auctions
Motivation
Under-utilized wireless spectrum
Provide incentive for primary users
General Framework
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment

5. Background Spectrum Opportunity Allocation Mechanism SUs’ bids Payment
Dynamic Spectrum Access Auctions
General Framework
Auction mechanisms
Periodic Auction [G. Kasbekar TON’10] , [A. Gopinathan INFOCOM’11] , [L. Chen]
Online Auction [H. Zheng INFOCOM’11] , [X. Li DySPAN’10] , [S. Sodagari JSAC’11]
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment

6. Background Spectrum Opportunity Allocation Mechanism SUs’ bids Payment
Dynamic Spectrum Access Auctions
General Framework
Auction mechanisms
Single Auction [S. Kasbeka TON’10]
Double Auction [W.Saad, INFCOM’10]
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment

7. Background This assumption doesn’t fit the real scenario Spectrum
Dynamic Spectrum Access Auctions
General Framework
Auction mechanisms
Periodic Auction
Online Auction
Single Auction
Double Auctino
Spectrum
Opportunity
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment
This assumption doesn’t fit the real scenario
Fixed and Homogenous
Fixed time span
Fixed bandwidth

8. Background Frequency Time Auction Period SU1:
Dynamic Spectrum Access Auctions
An example of SUs’ flexible requirements
Frequency
Time
Auction Period
SU1:
Spectrum Pool of Primary Operator
Rigid and inflexible spectrum offered by PUs
Cannot cater to the flexible requirements
Time-frequency flexible requirements
SU2:
SUs:
SU3:
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment

9. Objective DSA mechanism based on combinatorial auction
Consider the SUs’ requirements varying over time and frequency
Achieve efficiency, truthfulness & low computational complexity

10. Outline Introduction System Model and Problem Formulation System Model
Solution of CAP Under General Flexibility Model
Solution of CAP Under Modified Model
Simulation Results and Future Works 10

11. System Model Four-layer cognitive radio network model
Primary Operator based system

12. System Model
Four-layer cognitive radio network model
Periodic Auction

13. System Model Four-layer cognitive radio network model
Heterogeneous and flexible requirements

14. System Model 1 2 3 Model of PO (seller)’s spectrum opportunity
Time-frequency divide
We now denote the time interval as and the frequency interval as and every spectrum that starts from as
Because of the stable assumption, we change the notation of to ,where 1 2 3

15. System Model
Model of SUs(buyers)’ time-frequency flexible requirements
There are SUs denoted as
The bid of SU is denoted as , where and is the valuation of
The winning SU is denoted as
The payments of SUs are
The net utility of SU is:

16. Problem Formulation
Cognitive Radio Winning SUs Determination Problem(CRWDP):
For every subset , let denote the value of buyer for . We use to denote that the buyer wins the bundle and to denote the buyer loses or does not bid on . The winner determination problem is defined as follows:
Remarks:
Optimize the social welfare
One spectrum slot is assigned to one SU
One buyer can only have at most one bundle of goods at last
One buyer can only require one bundle of goods

17. Combinatorial Auction Problem (CAP)
Problem Formulation
The Truthful Mechanism Design Problem(TMDP):
For any buyer ,
Truthful bid: Truthful utility:
Declared bid: Untruthful utility:
Assume No constraint on
The TMDP problem is to design a payment mechanism such that
Combinatorial Auction Problem (CAP)
Winner Determination
Algorithm
Spectrum
Opportunity
Allocation
Mechanism
SUs’ bids
Payment
Truthful Payment
Mechanism

18. Outline Introduction System Model and Definitions
Solution of CAP Under General Flexibility Model
Approximation Analysis of the NP-hard CRWDP
Approximation Algorithm to Solve CRWDP
Truthful Payment Mechanism under the Approximation Algorithm
Solution of CAP Under Modified Model
Simulation Results and Future Works 18

19. Approximation Analysis of CRWDP
The CRWDP is NP hard to solve unless NP =ZPP
SUs: Vectors; Spectrum Slots: Edges; Values: all set to one
CRWDP is reduced from the maximum independent set problem(MISP)
The upper bound of the approximation ratio of CRWDP is , for any polynomial time algorithm.
The approximation ratio of CRWDP will not exceed that of MISP
The approximation ratio of MISP is
There is an implicated assumption in
the reduction:

20. Approximation Algorithm for CRWDP
Sorting Based Greedy Algorithm for CRWDP
Step 1: Reorder the bids according to a newly defined Norm
Step 2: Allocate spectrum opportunity greedily following the reordered list of norm

21. Approximation Algorithm for CRWDP
Sorting Based Greedy Algorithm for CRWDP
The choice of ordering norm is the critical
The computational complexity is
The approximation ratio of this algorithm reaches the upper bound

22. Omitted Approximation Algorithm for CRWDP
Proof of the approximation ratio of the proposed algorithm
Omitted

23. Truthful Payment Mechanism
The Truthful Payment Mechanism
Recall is the list in which all buyers are reordered by the norm in the first step of algorithm 1. And for buyer , we denote a buyer as the first buyer following in that has been denied but would have been granted were it not for the presence of . We have the following payment:
pays zero if his bid is denied or does not exist.
If there exists an and ’s bid is granted, he pays , where is the norm of

24. Truthful Payment Mechanism
The Proof of Truthfulness
If an auction mechanism fits the following conditions, it is truthful
Ex-post Budget Balance: That means the buyers are all rational so that they will not pay more than their value of the goods.
Monotonicity: A buyer who wins with can still win with any and any , when others’ bids are fixed.
Critical Payment: There exists a critical value for a winner , so that he only needs to pay this critical value to win. That is to say, if others’ bids are fixed, the payment of a certain winner does not depend on how he reports his bid

25. Outline Introduction System Model and Problem Formulation
Solution of CAP Under General Flexibility Model
Solution of CAP Under Modified Model
A Rational Modification of the System Model
Optimal Solution for CRWDP-C
Truthful Payment Mechanism under the Modified Model
Simulation Results and Future Works 25

26. A Rational Modification of the System Model
Additional Constraints on SUs’ Requirements
Full Time Usage
Consecutive Requirement
Change of Expressions in System Model
Goods Set: where is the starting frequency point of the th interval:
Redefine the as a close interval , which means the SU wants the frequency that starts from and ends at
to denote the highest valuation submitted to auctioneer on
G = {T|T ⊆ S and T fits the Consecutive
Requirement condition}. 26

27. A Rational Modification of the System Model
The problem of CRWDP-C can be formulated as:
there is one and only ,
for all other , 27

28. Optimal Solution for CRWDP-C
Algorithm achieves optimal solution for CRWDP-C and the computational complexity is 28

29. Payment Mechanism for the Modified Model
VCG-Based Truthful Payment Mechanism.
We denote the payment for buyer as . We denote as the goods buyer receives in the final allocation. We can see that when buyer wins and when loses. Further, we denote as valuation obtained by buyer in the final allocation. Note that when wins and when loses. We denote as the optimal social welfare obtained in the auction where is absent. Note that despite , all other buyers stay the same when calculating and We have that the payment is:
Remarks:
VCG-Payment Mechanism is truthful
VCG-Payment Mechanism’s limitation 29

30. Outline Introduction System Model and Problem Fromulation
Optimal Solution for CRWDP-C
Truthful Payment Mechanism under the Modified Model
Simulation Results and Future Works 30

31. Simulation Results and Future Works
Simulation Results under General Model

32. Simulation Results and Future Works
Simulation Results under Modified Model

33. Simulation Results and Future Works
Multiple bids can be submitted by SUs
Make the combinatorial auction mechanism an online one

34. Q&A

35. Thank you for listening
11/29/2018