Spectrum Bidding in Wireless Networks and Related. Ping Xu. Advisor: Xiang-Yang Li 1. Feb. 11th, 2008

Transcription

1 1 Advisor: Xiang-Yang Li 1 1 Illinois Institute of Technology Feb. 11th, 2008

2 Outline :

3 : Spectrum Scarcity Problem Figure: Frequency Allocations of The Radio Spectrum in US. university-log

4 White Space - Unused Spectrum To avoid interference Spectrum utilization with fixed allocation depends strongly on time and place. "In more congested areas, there is still ample space." 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 university-log

5 White Space - Unused Spectrum To avoid interference Spectrum utilization with fixed allocation depends strongly on time and place. "In more congested areas, there is still ample space." 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 university-log

6 White Space - Unused Spectrum To avoid interference Spectrum utilization with fixed allocation depends strongly on time and place. "In more congested areas, there is still ample space." 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 university-log

7 Opportunistic or Dynamic Spectrum Allocation Cognitive radio(cr) Spectrum auction How to deal with selfish behavior? Combine game theory with wireless communication modeling

8 Opportunistic or Dynamic Spectrum Allocation Cognitive radio(cr) Spectrum auction How to deal with selfish behavior? Combine game theory with wireless communication modeling

9 Opportunistic or Dynamic Spectrum Allocation Cognitive radio(cr) Spectrum auction How to deal with selfish behavior? Combine game theory with wireless communication modeling

10 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

11 Network Model Primary user U who holds the right of some spectrum channels secondary users V = {v 1, v 2,, v n } who wants to lease the right of some spectrum channels in some geometry region D(vi, r i ) for some time period Ti for some frequencies Fi with bid bi private valuation wi

12 Network Model Primary user U who holds the right of some spectrum channels secondary users V = {v 1, v 2,, v n } who wants to lease the right of some spectrum channels in some geometry region D(vi, r i ) for some time period Ti for some frequencies Fi with bid bi private valuation wi

13 Network Model A direct viewing graph for a single channel Figure: An illustration of cylinder graph

14 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.

15 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)

16 Example For example, problem SUI represents Channel requirement: Single-minded Region requirement: Unit Disks Time requirement: Time Interval

17 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.

18 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

19 Problem YOM Channel requirement: Y(Single channel) Region requirement: Overlap Time requirement: Mixed(Time interval or duration)

20 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.

21 Problem SUI Channel requirement: Single-minded Region requirement: Unit disk Time requirement: Time Interval

22 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: S 1 = {1, 2, 5}, w(s 1 )=10 S 2 = {2, 4, 5}, w(s 2 )=15 S 3 = {3, 4}, w(s 3 )=10 Find the maximum weighted subset {S1, S 3 } 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

23 Algorithm for Problem SUI Upper Bound for Problem SUI Hastad [1] proved that Set Packing cannot be approximated within m 1/2 approximation unless NP=ZPP, where m is the size of universal set. Therefore, the upper bound for problem SUI is O( m) approximation.

24 Algorithm for Problem SUI Greedy algorithm for Set Packing doesn t work Two kinds of greedy algorithm can achieve m 1/2 approximation for set packing problem. Sort subsets by descending w(s i ) - proposed by Lehmann Si et al. [2]. Sort subsets by nondecreasing order of w(n i ) w(s i ), where w(n i) is the total weight of subsets which intersect subset S i - proposed by Sakai et al. [3]. Neither greedy algorithm works for problem SUI.

25 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.

26 Algorithm for Problem SUI Approximation algorithm for G 1 Find optimal solution OPT i for each single channel i. Use max{opt 1, OPT 2,, OPT m } as solution. Since each bidder bids at least m frequencies, m OPT i mopt(g 1 ) i=1 m m max OPT i i=1 m OPT(G 1) = 1 OPT(G 1 ). m

27 Algorithm for Problem SUI Approximation algorithm for G 2 Convert this problem into Scheduling Split Intervals Problem (SSIP). A set of weighted jobs: Job 1, w(job1)=10 Job 2, w(job2)=20 Job 3, w(job3)=15 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. university-log

28 Algorithm for Problem SUI Convert SUI to SSIP User1 requests F1, F2 User2 requests F2, F3 User3 requests F1, F3 Time Period T Figure: An instance for Problem SUI

29 Algorithm for Problem SUI Convert SUI to SSIP User1 requests F1, F2 User2 requests F2, F3 User3 requests F1, F3 Time Period T T T Figure: An corresponding instance for SSIP Since SSIP is 2t-approximattable, we get 2 m-approximation solution. university-log

30 Algorithm for Problem SUI Approximation ratio The maximum of the above two solutions is at least ( OPT(G1 ) max, OPT(G ) 2) m 2 1 m 3 m OPT, Approximation ratio 2 4 can be achieved when partition m the groups with at least m 2 frequencies.

31 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).

32 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.

33 Summary 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.

34 Future works Spectrum auction problems, such as problem YOM. Apply game theory to Cognitive radio(cr)

35 References 1 J. Hastad. Clique is hard to approximate within n 1 ε, Acta Mathematica. 1999, Volume 182, pp 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 Shuichi Sakai and Mitsunori Togasaki and Koichi Yamazaki. A note on greedy algorithms for the maximum weighted independent set problem, Discrete Appl. Math , Volume 126, pp 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 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 university-log

36 Questions and Comments?

37 Algorithm for Problem SUI Counterexamples for the first greedy algorithm which sorts the b bidders by descending i. Fi d i Only two bidders, i and j b i b j b i Fi d i < b j Fj d j The total weight we get is b j, that is arbitrarily smaller than b i.

38 Algorithm for Problem SUI Counterexamples for the second greedy algorithm which sorts the bidders by nondecreasing order of b(n i) b i. The figure gives an instance such that the solution is at most O( 1 n ) of the optimal one = time interval bidded by i 111 = time intervals bidded by Ni Figure: This is a view from the lateral face(xy -axis).