Spectrum Bidding in Wireless Networks and Related Ping Xu 1 Advisor: - - PowerPoint PPT Presentation

university-logo Spectrum Bidding in Wireless Networks and Related

Spectrum Bidding in Wireless Networks and Related Ping Xu1 Advisor: Xiang-Yang Li1

1Illinois Institute of Technology

• Feb. 11th, 2008

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related

Outline

Spectrum Bidding in Wireless Networks and Related:

◮ Background ◮ Problem Formulation ◮ Our Approaches ◮ Summary and Future Works

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background: Spectrum Scarcity Problem

Figure: Frequency Allocations of The Radio Spectrum in US.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

White Space - Unused Spectrum

◮ To avoid interference ◮ Spectrum utilization with fixed allocation depends strongly

• n time and place. "In more congested areas, there is still

ample space." – www.tvtechnology.com

◮ Dallas – 40 percent ◮ Boston – 38 percent ◮ Seattle – 52 percent ◮ San Francisco – 37 percent.

◮ A result of technical changes. For example, the planned

switchover to digital television may free up large areas between 54MHz and 698MHz.

◮ "Battle Heats Up for TV Spectrum "White Space" Use" –

WIMAX.com

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

White Space - Unused Spectrum

◮ To avoid interference ◮ Spectrum utilization with fixed allocation depends strongly

• n time and place. "In more congested areas, there is still

ample space." – www.tvtechnology.com

◮ Dallas – 40 percent ◮ Boston – 38 percent ◮ Seattle – 52 percent ◮ San Francisco – 37 percent.

◮ A result of technical changes. For example, the planned

switchover to digital television may free up large areas between 54MHz and 698MHz.

◮ "Battle Heats Up for TV Spectrum "White Space" Use" –

WIMAX.com

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

White Space - Unused Spectrum

◮ To avoid interference ◮ Spectrum utilization with fixed allocation depends strongly

• n time and place. "In more congested areas, there is still

ample space." – www.tvtechnology.com

◮ Dallas – 40 percent ◮ Boston – 38 percent ◮ Seattle – 52 percent ◮ San Francisco – 37 percent.

◮ A result of technical changes. For example, the planned

switchover to digital television may free up large areas between 54MHz and 698MHz.

◮ "Battle Heats Up for TV Spectrum "White Space" Use" –

WIMAX.com

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

Opportunistic or Dynamic Spectrum Allocation

◮ Cognitive radio(CR) ◮ Spectrum auction

◮ How to deal with selfish behavior? ◮ Combine game theory with wireless communication

modeling

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

Opportunistic or Dynamic Spectrum Allocation

◮ Cognitive radio(CR) ◮ Spectrum auction

◮ How to deal with selfish behavior? ◮ Combine game theory with wireless communication

modeling

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Background

Opportunistic or Dynamic Spectrum Allocation

◮ Cognitive radio(CR) ◮ Spectrum auction

◮ How to deal with selfish behavior? ◮ Combine game theory with wireless communication

modeling

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Main Idea

Construct an auction to assign spectrum

◮ Auctioneer: primary users ◮ Bidders: secondary users, selfish, but rational ◮ Objects

◮ Determine winners and payments ◮ Maximize the social efficiency - total valuation of winners Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Network Model

◮ Primary user U who holds the right of some spectrum

channels

◮ secondary users V = {v1, v2, · · · , vn} who wants to lease

the right of some spectrum channels

◮ in some geometry region D(vi, ri) ◮ for some time period Ti ◮ for some frequencies Fi ◮ with bid bi ◮ private valuation wi Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Network Model

◮ Primary user U who holds the right of some spectrum

channels

◮ secondary users V = {v1, v2, · · · , vn} who wants to lease

the right of some spectrum channels

◮ in some geometry region D(vi, ri) ◮ for some time period Ti ◮ for some frequencies Fi ◮ with bid bi ◮ private valuation wi Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Network Model

A direct viewing graph for a single channel

Figure: An illustration of cylinder graph

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Object

Find an allocation method, which must be

◮ conflict free in geometry region, time period and

frequencies

◮ maximize the social efficiency - total valuation of winners.

In most cases, this problem is NP-hard.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Problem Formulation

For notational convenience, we use CRT to denote a version of problem under special assumption, where

◮ Channel requirement

◮ S(single-minded), F(flexible-minded), Y(single channel)

◮ Region requirement

◮ O(overlap), U(unit disks), G(general disks)

◮ Time requirement

◮ I(time interval), D(time duration), M(time interval or

duration)

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Example

For example, problem SUI represents

◮ Channel requirement: Single-minded ◮ Region requirement: Unit Disks ◮ Time requirement: Time Interval

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Some well-known problems

◮ Knapsack problem: Problem YOD. ◮ Set packing problem: a special case of problem SOI, SUI,

SGI.

◮ Maximum weighted independent set problem of a disk

graph: a special case of Problem YGI

◮ Multi-knapsack problem: a special case of Problem YUD.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Our Results

Problems we mainly focus on

◮ problem YOM: 1/2 approximation algorithm ◮ problem YUI: PTAS ◮ problem YUD: 1/9 approximation algorithm ◮ problem YUM: 1/10 approximation algorithm ◮ problem SUI: Θ(√m) approximation algorithm

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Problem YOM

◮ Channel requirement: Y(Single channel) ◮ Region requirement: Overlap ◮ Time requirement: Mixed(Time interval or duration)

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem YOM

◮ Partition bidders into two groups according to their time

requirements.

◮ Find the best solution F to the group require time intervals

with dynamic programming.

◮ Find the approximated best solution S′ to the group with

the FPTAS for knapsack problem.

◮ max(F, S′) ≥ 1 2(1+ǫ) · OPT.

So we have a simple 2 + ǫ′ approximation algorithm.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Problem SUI

◮ Channel requirement: Single-minded ◮ Region requirement: Unit disk ◮ Time requirement: Time Interval

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Upper Bound for Problem SUI

◮ Set packing problem is a special case of the problem SUI.

◮ A universal Set E = {1, 2, 3, 4, 5} ◮ A set of weighted subsets: ◮ S1 = {1, 2, 5}, w(S1)=10 ◮ S2 = {2, 4, 5}, w(S2)=15 ◮ S3 = {3, 4}, w(S3)=10 ◮ Find the maximum weighted subset {S1, S3} which is

conflict free.

◮ Set packing is a special case of the problem SUI where

each user conflicts with each other in region and time requirement

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Upper Bound for Problem SUI

◮ Hastad [1] proved that Set Packing cannot be

approximated within m1/2 approximation unless NP=ZPP , where m is the size of universal set.

◮ Therefore, the upper bound for problem SUI is O(√m)

approximation.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Greedy algorithm for Set Packing doesn’t work

◮ Two kinds of greedy algorithm can achieve m1/2

approximation for set packing problem.

◮ Sort subsets by descending w(Si)

√

|Si| - proposed by Lehmann

et al. [2].

◮ Sort subsets by nondecreasing order of w(Ni)

w(Si), where w(Ni)

is the total weight of subsets which intersect subset Si - proposed by Sakai et al. [3].

◮ Neither greedy algorithm works for problem SUI.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Main idea for problem SUI

◮ Partition the bidders into two groups:

◮ G1 contains all the bidders that request at least √m

frequencies.

◮ G2 contains all the other bidders, i.e., request less than √m

frequencies.

◮ Solve each group with Θ(√m) approximation algorithm.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Approximation algorithm for G1

◮ Find optimal solution OPTi for each single channel i. ◮ Use max{OPT1, OPT2, · · · , OPTm} as solution. ◮ Since each bidder bids at least √m frequencies, m

• i=1

OPTi ≥ √ mOPT(G1)

m

max

i=1 OPTi ≥

√m m OPT(G1) = 1 √mOPT(G1).

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Approximation algorithm for G2 Convert this problem into Scheduling Split Intervals Problem (SSIP).

◮ A set of weighted jobs: Job 3, w(Job3)=15 Job 1, w(Job1)=10 Job 2, w(Job2)=20

Figure: A simple example of SSIP

◮ Find a conflict free subset with maximum total weight.

BAR-YEHUDA et al. [4] proved that SSIP is 2t-approximatable.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Convert SUI to SSIP Time Period User2 requests F2, F3 User1 requests F1, F2 User3 requests F1, F3

T

Figure: An instance for Problem SUI

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Convert SUI to SSIP

Time Period User1 requests F1, F2 User3 requests F1, F3 User2 requests F2, F3

T T T

Figure: An corresponding instance for SSIP

Since SSIP is 2t-approximattable, we get 2√m-approximation solution.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Approximation ratio

◮ The maximum of the above two solutions is at least

max OPT(G1) √m , OPT(G2) 2√m

• ≥

1 3√mOPT,

◮ Approximation ratio √ 2 4√m can be achieved when partition

the groups with at least

• m

2 frequencies.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Approximation ratio Θ(t) for SSIP is asymptotically tight.

◮ If there is approximation algorithm with ratio o(t), using the

similar trick above, we will have an algorithm for SUI (and thus set packing problem) with approximation ratio o(m

1 2 ). Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Truthful Mechanism Design

◮ Monotone output algorithm

◮ Almost all algorithms above are monotone. However, the

classic FPTAS for knapsack problem is not monotone. Patrick Briest [5] proposed a monotone FPTAS for knapsack problem.

◮ Critical value payment scheme

◮ A critical value payment scheme charges winners critical

value, otherwise, does not charge anything.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Summary

Spectrum Bidding in Wireless Networks and Related

◮ Propose the problem: combine game theory with

communication modeling to solve the problem of selfish behavior.

◮ Formulate several versions of problems by separately

assuming the frequency, region and time requirement

◮ Design approximation algorithm or PTAS for those

problems.

◮ Design truthful mechanism which maximizes the society

efficiency.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Future works

◮ Spectrum auction problems, such as problem YOM. ◮ Apply game theory to Cognitive radio(CR)

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

References

1 J. Hastad. “Clique is hard to approximate within n1−ε”, Acta

• Mathematica. 1999, Volume 182, pp. 105–142

2 Daniel J. Lehmann and Liaden Ita O’Callaghan and Yoav Shoham. “Truth revelation in approximately efficient combinatorial auctions”, ACM Conference on Electronic Commerce. 1999, pp. 96–102 3 Shuichi Sakai and Mitsunori Togasaki and Koichi Yamazaki. “A note on greedy algorithms for the maximum weighted independent set problem”, Discrete Appl. Math.. 2003, Volume 126, pp. 313–322 4 Reuven Bar-Yehuda and Magnús M. Halldórsson and Joseph (Seffi) Naor and Hadas Shachnai and Irina Shapira. “Scheduling split intervals”, SODA ’02: Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms. 2002, pp. 732–741 5 Patrick Briest and Piotr Krysta and Berthold Vöcking. “Approximation techniques for utilitarian mechanism design”, STOC ’05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing. 2005, pp. 39–48

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Questions and Comments?

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Counterexamples for the first greedy algorithm which sorts the bidders by descending

bi

√

|Fi|×di . ◮ Only two bidders, i and j ◮ bi ≫ bj ◮ bi

√

|Fi|×di < bj

√

|Fj|×dj

The total weight we get is bj, that is arbitrarily smaller than bi.

Ping Xu Spectrum Bidding in Wireless Networks and Related

university-logo Spectrum Bidding in Wireless Networks and Related Background Problem Formulation Our Approaches Summary and Future Works

Algorithm for Problem SUI

Counterexamples for the second greedy algorithm which sorts the bidders by nondecreasing order of b(Ni)

bi .

The figure gives an instance such that the solution is at most O( 1

√n) of the optimal one.

= time interval bidded by i

• = time intervals bidded by Ni

Figure: This is a view from the lateral face(XY-axis).

Ping Xu Spectrum Bidding in Wireless Networks and Related