combinatorial auction with time-frequency flexibility in cognitive radio networks

04/22/2023 1

Combinatorial Auction with Time-FrequencyFlexibility in Cognitive Radio Networks

Mo Dong*, Gaofei Sun*, Xinbing Wang*, Qian Zhang**

*Department of Electronic EngineeringShanghai Jiao Tong University, China

**Department of Comp. Sci. and EnginHK Univ. of Sci. and Tech., HongKong

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

BackgroundObjectives

System Model and Problem Formulation

Solution of CAP Under General Flexibility

Model

Solution of CAP Under Modified Model

Simulation Results and Future Works

3

Background Dynamic Spectrum Access Auctions

Motivation

• Under-utilized wireless spectrum

• Provide incentive for primary users

General Framework

4

Spectrum

Opportunity

Allocation

Mechanism

SUs’ bids

Payment

Mechanism

5

Background 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

Mechanism

6

Background 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

Mechanism

7

Background Dynamic Spectrum Access Auctions

General Framework

Auction mechanisms

• Periodic Auction

• Online Auction

• Single Auction

• Double Auctino

Spectrum

Opportunity

Allocation

Mechanism

SUs’ bids

Payment

Mechanism

Spectrum

Opportunity

Fixed and HomogenousFixed time span

Fixed bandwidth

This assumption doesn’t fit the real scenario

8

Background Dynamic Spectrum Access Auctions

An example of SUs’ flexible requirements

SU1:

SU2:

SU3:

Frequency

TimeAuction Period

Rigid and inflexible spectrum offered by PUs

Cannot cater to the flexible requirementsTime-frequency flexible requirements

Spectrum Pool of Primary Operator

Spectrum

Opportunity

Allocation

Mechanism

SUs’ bids

Payment

Mechanism

SUs:

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

Problem Formulation

Solution of CAP Under General Flexibility

Model

Solution of CAP Under Modified Model

Simulation Results and Future Works10

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

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 .

S

tf t f 0 0( , )t f 0 0( , )t fs

0 0( , )t fsis {1,...., }i m

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:

N {1,..., }U Ni ( , )i i ib C v iC S iv

iC

1{ ,..., }kW w w

1 2{ , ,...., }nP p p p

i

if wins the auction0 if loses the auctioni i

i

v p iu

i

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

T S ( )jb T j U T( , ) 1e T j j T

( , ) 0e T j

max ( ) ( , ) j

j U T S

b T e T j

. . ( , ) 1, i

iT s j U

s t e T j s S

( , ) 1, T S

e T j j U

* * *, s.t. ( ) 0 and , ( ) 0i ii i ii U T S b T T T b T

T

17

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

i U* * *{ , }i i ib C v

{ , }i i ib C v*

i iC C iv

* * *( ) ( )i i i i iu b v p b *( ) ( )i i i i iu b v p b

* *( ) ( ), , Ci i i i i i iu b u b v R C

Spectrum

Opportunity

Allocation

Mechanism

SUs’ bids

Payment

Mechanism

Winner Determination

Algorithm

Truthful Payment

Mechanism

Combinatorial Auction Problem (CAP)

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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 Works18

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:

m

1n

2m n

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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

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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

O( log( ))n n

m

22

Approximation Algorithm for CRWDP

Proof of the approximation ratio of the proposed algorithm

Omitted

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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 .

Li L

( )l i i Li

i ( )l i

( )l i i ( )*i l ic n( )l i( )l in

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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

* *{ , }b C v' *v v*'C C

c iv vi

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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

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}.

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1 2{ , ,...., }mS f f f ifi [ , ]i if f f

iC [ , ]iC r qrf qf

([ , ])i jh f f[ , ]i jf f

A Rational Modification of the System Model

The problem of CRWDP-C can be formulated as:

there is one and only ,

for all other , .

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max ( ) TT G

h T e

T

s.t. 1, i

T if

e f S

{0,1} Te T G

i U * * s.t. ( ) 0i i iT S b T *iT T ( ) 0ib T

Optimal Solution for CRWDP-C

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

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2O( )n

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

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i ip igi i ig C

i ig i ( )iv gi ( )i iv g v

i ( ) 0iv g i opt iv

ii optv

opt iv

( )i opt i i optp v v g v

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Outline Introduction System Model and Problem Fromulation Optimal Solution for CRWDP-C Truthful Payment Mechanism under the

Modified Model Simulation Results and Future Works

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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

Future Works

Multiple bids can be submitted by SUs

Make the combinatorial auction mechanism an online one

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Q&A

04/22/2023 35

Thank you for listening