Cost Sharing and Approximation成本分担与逼近共34页

•Exact cross-monotonic sharing exists if c*() submodular
[Moulin&Shenker 98]
•Exact cost sharing for spanning tree
[Kent&Skorin-Kapov 96], [Jain&Vazirani 01]
Cost sharing fn: ξ : 2UU + ξ(S,i) – share of user i, if set S served
need to recompute tree & payments!
cross-monotonic: for iSTU
ξ(U, ) ≤ ξ(U-, )
ξ(S,i) ≥ ξ(T,i)
user shares cost
Competitiveness: ΣiS pi ≤ c*(S)
for any SU: no overpayment
Known as core in game theory often empty for games of interest
p+p+p+p+p ≥ c( ) p+p+p ≤ c( )
ui
cost share
Theorem: [Moulin&Shenker]: ξ(.) cross-monotonic mechanism group strategyproof
14 Oct 03
Cost sharing & Approximation
6
Known cross-monotonic functions
14 Oct 03
Cost sharing & Approximation
5
The Moulin&Shenker mechanism
S := U repeat
ask each user i “is ξ(S,i) ≤ ui ?”
drop all iS who say NO until all iS say YES Output: set S; prices pi = ξ(S,i)
U = {,,,,}
c*(S) – cost of optimal infrastructure for users in S often NP-hard to compute
14 Oct 03
Cost sharing & Approximation
3
Sharing the cost
Approximate Cost Recovery: ΣiU pi ≥ c*(U)
constructive proof givesБайду номын сангаасan approximation algorithm
14 Oct 03
Cost sharing & Approximation
8
Approximation algorithms
•Primal-dual approximation for Facility Location
[Jain&Vazirani99] , [Mahdian,Ye&Zhang 02], [Mettu&Plaxton 00]
•Approximation for Single Sink Rent or Buy
[Karger&Minkoff 00], [Meyerson&Munagala 00] [Swamy&Kumar 02], [Gupta,Kumar&Roughgarden 03]
14 Oct 03
Cost sharing & Approximation
4
Modeling the users
ui – value of service for i User may say NO if pi>ui util(i) = ui – pi if i gets service
0 otherwise
Used our method to construct a cost sharing fn for:
Metric Facility Location Single Sink Rent or Buy
recovers 1/3 of cost recovers 1/15 of cost
competitive, cross-monotonic
Cost Sharing and Approximation成本 分担与逼近
服从真理，就能征服一切事物
Group strategy proof mechanisms via primal-dual algorithms
(Cost Sharing)
Martin Pál
É va Tardos
14 Oct 03
implies 2-approx. cost sharing for Steiner tree
Any other games for which cross-mono sharing exists?
14 Oct 03
Cost sharing & Approximation
7
Our results
General method for generating competitive, crossmonotonic cost shares using primal-dual algorithms
14 Oct 03
Cost sharing & Approximation
9
Single Sink Rent or Buy
U is a set of users. r is the sink (root) node. graph G with edge lengths ce.
1) Find a path from each user to sink.
2) Rent or Buy each edge.
Rent: pay ce for each path using e
Buy: pay M ce Goal: minimize
Cost sharing & Approximation
2
Our setting
Universe U of (selfish) users Users want to benefit from shared infrastructure cost function c*()
c*(U) = cost of c*() = cost of