Sharing the cost of multicast transmissions in wireless networks
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Transcript of Sharing the cost of multicast transmissions in wireless networks
Sharing the cost of multicast transmissions in wireless networksCarmine Ventre
Joint work with Paolo Penna
University of Salerno, WP2
Wireless transmission
Power(i)= d(i,j)α = range(i) α, α>1 (empty space α = 2)
A message sent by station i to j can be also received by every station in transmission range of ii
j
d(i,j)α
Wireless multicast transmission
Who receives Roma-Juventus How to transmit Goal: maximize
Benefit – Cost i.e. the social welfare
Paolo 1€
10€ 1€ 1€ 3€
Carmine 1€ Christos 10€ Andrea 30€
Pino 50€
known
private
source
Selfish agents
COST = 10 + 5 = 15 WORTH = 50 + 30 = 80 NET WORTH = 80 – 15 =
65
source
10
10
5
Pino 50 €
Andrea 30 €
Paolo 9 €
0 €
Pino says 0 € and gets
Roma – Juventus
for free
5.1 €Andrea says 5.1 € and gets
Roma – Juventus
for a lower price
Andrea says 5.1 €
Pino says 0 €Nobody gets
Roma - Juventus
NW’ = 0
WYSWYP (What You Say What You Pay)
Graph model
A complete directed weighted communication graph G=(S,E,w)
w(i,j) = cost of link (i,j) w(1,4) = d(1,4)2.1
w(1,2) = d(1,2)5
w(2,4) = ∞ w(4,2) = d(4,2)2.1
A source node s vi = private valuation of
agent i
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v1 v2
Mechanism design: model
Design a mechanism M=(A,P) Each agent declares bi
Algorithm A selects, based on (b1, …, bn), a set of receivers a subset of connection T E
Agent i must pay Pi(b1, …, bi-1, bi, bi+1 ..., bn) Utility of the agent
ui(bi)=
Goal of agent i: maximize ui(bi)
otherwise.0
ion, transmiss thereceives i if)b,...,b,...,(bPv ni1ii
Mechanism’s desired properties No positive transfer (NPT)
Payments are nonnegative: Pi 0
Voluntary Participation (VP) User i is charged less then his reported valuation
bi (i.e. bi ≥ Pi)
Consumer Sovereignty (CS) Each user can receive the transmission if he is
willing to pay a high price.
Mechanism’s desired properties: Incentive Compatibility Strategyproof (truthful) mechanism
Telling the true vi is a dominant strategy for any agent
Group-strategyproof mechanism No coalition of agents has an incentive to jointly
misreport their true vi
Stronger form of Incentive Compatibility.
Mechanism’s desired properties Budget Balance (BB)
Pi = COST(T) (where T is the solution set)
Efficiency (NW) the mechanism should maximize the
NET WORTH(T) := WORTH(T)-COST(T)
where WORTH(T):= iT vj
Mutually exclusive!!
Efficiency No Group strategy-proof
Previous work
Wireless broadcast 1d: COSTopt in polynomial time [Clementi et al, to appear] 2d: NP-hard, MST is an O(1)-apx [Clementi et al, ‘01] On graphs: (log n)-apx [Guha et al ‘96, Caragiannis et al, ‘02] Many others…
Wired cost sharing (selfish receivers) Distributed polytime truthful, efficient, NPT, VP, and CS mechanism
for trees (no BB) [Feigenbaum et al, ‘99] Budget balance, NPT, VP, CS and group strategy-proof mechanism
(no efficiency) [Jain et al, ‘00] No α-efficiency and β-BB for each α, β > 1 [Feigenbaum et al, ‘02] polytime algorithm no R-efficiency, for each R > 1 [Feigenbaum et
al, ‘99]
Our results
G is a tree NWopt in polytime distributed algorithm Polytime mechanism M=(A,P) truthful, NPT, VP and CS Extensions to “metric trees” graphs
G is not a tree 2d: NP-hard to compute NWopt
1d: Polytime mechanism M=(A,P) truthful, NPT, VP, CS and efficient (i.e. NW is maximized)
Precompute an universal multicast tree T G A polytime truthful, NPT, VP and CS mechanism O(1) or O(n)-efficiency, in some cases
polytime algorithm no R-efficiency, for every R > 1
VCG Trick (marginal cost mechanism) Utilitarian problem:
Xsol, measure(X)=i valuationi(X)
Aopt computes X sol maximizing measure(X)
PVCG: M=(Aopt, PVCG) is truthful
VCG Trick (marginal cost mechanism)Making our problem utilitarian:
measure(X) valuationi(X)
WORTH(X)-COST(X)
= i
iXvi = WORTH(X)
vi
ciInitially, charge to every receiver ithe cost ci of its ingoing connection
- ci
- COST(X)
Pi = ci + PVCG
Free edges on Trees
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s
graphtree
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5
s
RECURSION?
NO! YES!
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4 5 4 5
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Trees algorithm: recursive equation
jk ccchkoptj
chjopt kNWcNW
),i()i(
i )(max,0maxv)i(
It is easy to see that the best solution has an optimal substructure
It is simple to compute NWopt(s) in distributed bottom-up fashion
O(n) time, 2 msgs per link
k s.t. ck ≤ cj
i
j
cj
vi
Trees with metric free edges
Path(i,4)=w(i,1)+w(1,4) w(i,3) ≥ path(i,4)
(i,4) metric free edge
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5
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1 5
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Tree with metric free edge: idea A node k reached for free gets some credit
i
j
cj
k gets cj-ck units of credit
k
ck
Tree with metric free edge: credit usage k can use its credit to
reach all of its children If there is a child l s.t. cl >
credit(k) and NWopt(l)>0 then credit(k) is useless For each r Є ch(k):
cl – cr > credit(k) – cr
Paying a free edge is not a good solution (i.e. we have a smallest credit and a greater cost)
k
k
lr
credit(r)=cl-cr
r
credit(r) = credit(k)-cr
credit(l)=0
Tree with metric free edge: recursive equations We have two contributions:
the nodes whose ingoing edge is paid
the nodes with credit c whose ingoing edge is free
)cc,k(NWc)(NW kicc),)(p(chk
optipay
ik
i
i
)(max,),(maxv)ci,(),(
),i(i jNWccjNWNW pay
ccichjccchj
joptoptj
j
NOTE: the optimum is NWopt(s,0)
The one dimensional Euclidean case Stations located on a line (linear network)
si j1 n
receivers
Clementi et al algo
(Some) Open problems
2d Euclidean case: O(1)-APX multicast algorithm “Good” universal Euclidean multicast trees Truthful mechanism with O(1)-APX BB truthful mechanisms