Cross-Monotonic Multicast Zongpeng Li Department of Computer - - PowerPoint PPT Presentation

Cross-Monotonic Multicast

Zongpeng Li Department of Computer Science University of Calgary April 17, 2008

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Multicast

• Multicast models one-to-many data dissemination in a

computer network

• Example: live Video Streaming on the Internet

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Min-Cost Multicast

• Network: G = (V, E)
• Link capacities: c : E → Q+
• Unit flow cost on each link: w : E → Q+
• Each receiver should receive data at rate d
• Flow rate on each link during routing: f : E → Q+
• Total routing cost:
• e∈E w(e)f(e)
• Goal: compute min-cost multicast flow f ∗ that achieves

multicast rate d

3

Min-Cost Multicast

Using multicast trees:

• w(e) = 1, ∀e (every tree link has a cost 1.0 for a unit

flow on it)

• For throughput d = 1: minimum tree has cost 5
• Q: how to share the cost among the receivers?

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Min-Cost Multicast

with network coding:

a a a b b b a+b a+b a+b

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

• Every link flow has rate 0.5.
• Total cost
• e w(e)f(e) = 0.5 × 9 = 4.5.
• Q: how to share the cost among the receivers?

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Min-Cost Multicast

• A multicast rate d is feasible in a directed network

if and only if it is feasible as a unicast rate to every multicast receiver independently. [Ahlswede et al., IT 2000]

• Optimal multicast can be modelled using LP [Lun et al.,

INFOCOM 2005][Li et al., INFOCOM 2005]

– Network flow LPs with extra constraints – Tailored solution algorithms, efficient, distributed Now: cooperative environment − → selfish/strategic re- ceivers

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Game Theory Aspects of Computer Networks

• Classic network protocol design assume cooperative

and altruistic user behavior – e.g., TCP congestion control

• Not always safe
• Network game theory — face the reality:

– strategic network users – selfish network traffic – market-driven network infrastructure

• Network operators, protocol designers : induce desired

behaviors from selfish network agents, for the well-being

• f the entire network

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The Multicast Game

• Selfish Traffic

– [Li, IEEE INFOCOM 2007] – Shadow prices from dual LP – Shadow price based costing sharing and link tolls – Every min-cost multicast flow can be enforced

• Selfish Users

– [Li, IEEE INFOCOM 2008] ← − You are here!

• Selfish Links

– Induce truthful cost reports from links – Apply the Vickrey-Clarke-Groves Mechanism – Efficient computation of Vickrey prices, ongoing work

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The Background Story

• Potential set of users, T (think of Internet media

streaming)

• Each with private valuation of the multicast service
• For each subset of potential users A ⊆ T

– Compute multicast routing fA – fA has cost |fA| – Share the cost among users in A

• Which set A to serve? How to share the cost?

– We really want users to tell us their true valuations!

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Strategyproof Multicast

• Strategyproof mechanisms

– Telling the truth is for the best interest of a user herself – dominant strategy

• Group-strategyproof mechanisms

– Further being robust against collusion

• The key: Cross-Monotonic Cost Sharing [Moulin,Shenker,

2001]

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Cross-Monotonicity

A B AB ABC ABE

. . . . . .

AC

Travel down any of these Criss-Crossing routes, a node’s cost share should be monotonically decreasing.

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From C-S to Group-stragetyproofness

The simultaneous Cournot Tatonnement [Moulin, 1982]:

• 1. Start with full set T
• 2. Ask each user: how much do you wish to pay for the

service?

• 3. Compute C-S cost share for each user
• 4. Exclude a user from the service set if her willingness to

pay is under the computed cost share

• 5. Loop to 2., till convergence
• 6. Serve users in the converged set

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Cross-Monotonic Multicast

• Cross-Monotonic

– √

• Optimal

– √ min-cost multicast routing

• Budget Balanced

– × √ Recover routing cost from user payments

• In-Core

– √ Users motivated to participate

• No-Positive Transfers

– √ Never pay someone to participate

• Efficient

– × Maximize net social utility

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The hardness of the problem

OPT + C-S + B-B = hard

• Submodular costs always have C-S sharing schemes
• Multicast cost is not submodular
• Fundamental conflict between primal optimality and

dual smoothness

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The hardness of the problem

OPT + C-S + B-B = hard

• OPT + C-S = easy

– Use optimal routing computed by LP – Every user always pays 0

• OPT + B-B = easy

– Use optimal routing computed by LP – Split link flow cost evenly among receivers using it

• C-S + B-B = easy

– Restrict route selection to one base tree – For each subset of users, use a corresponding subtree – Split link flow cost evenly among receivers using it

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The hardness of the problem

OPT + C-S + B-B = hard

• Direct LP dual is not smooth
• Local sharing is not in-core

S

T1 T2

4 4 3 S

T1 T2

4 3

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Directed Networks, positive result

A 1

k-budget-balanced, optimal, cross-monotonic multicast

scheme

For any A ⊆ T: (1) Solve min-cost multicast LP, let f ∗

A be the optimal solution

(2) for each receiver u in A: Solve min-cost S→u unicast flow f ∗

u

(3) Route multicast flows as specified in f ∗

A

(4) Let each u ∈ A pay yA(u) = |f∗

u|

|A| 17

Directed Networks, negative result

• No optimal, cross-monotonic, ( 2

√ k + ǫ)-budget-balanced

scheme, ∀ǫ > 0 ...

T11 T12

... ... ...

T22 T21 Tl

l-partite h n

• d

e s i n e a c h p a r t i t e

...

S

1

T1h T2h Tlh T1j 1 T2j 2 δ δ δ

2 l1

T

∀(j1, j2, ..jl) ∈ [1..h]l : Tljl

k: total number of potential multicast receivers k = hl

18

Directed Networks, negative result

Probabilistic proof

• randomly pick service set A = A1 ∪ A2, show expected

B-B factor is low

• A1: uniformly randomly pick a partite i
• A2: ∀j = i, uniformly randomly pick a user in partite j

EA(

Tij∈A yA(Tij))

=1 EA(

Tij∈A1 yA(Tij)) + EA( Tij∈A2 yA(Tij))

≤2 hEA2,Tij∈A1(yA2+Tij(Tij)) +EA2,Tij∈A1(

Tij∈A2 yA2+Tij(Tij))

=3 h( 1

l + δ) + (l − 1)( 1 l + δ)

=4 (h + l − 1)( 1

l + δ) ≤ 2 √ k 19

Undirected Networks, Negative Result

• No optimal, cross-monotonic, (1

2 + ǫ)-budget-balanced

scheme, ∀ǫ > 0

...

S

1 1 1 1

T1j 1 T2j 2 Tlj l

∀(j1, j2, ..jl) ∈ [1..h]l :

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Undirected Networks, Positive Result

• A

k+1 (2k+1)ζ-budget-balanced, optimal, cross-monotonic

multicast scheme

• ζ: the coding advantage

– proven: ≤ 2 [Li et al., CISS 2004] – contrived networks: ≤ 8

7

– random networks: always 1 – believed: always close to 1

• k+1

(2k+1)ζ should be close to 1 2, almost tight bound

• Idea: smooth dual growing, from primal-dual algorithm

design

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The Complexity for Maximum Budget-Balance

• Input: multicast network

– network topology – link capacities and costs – sender, potential receivers

• Output: maximum b-b ratio for optimal and cross-

monotonic multicast schemes

• Brute-force solution: two-stage linear optimization

(solve large # of LPs)

• NP-Hard?

22

Maximum B-B: two-stage linear optimization

Stage 1: ∀A ⊆ T, compute f ∗

A:

Minimize

• →

uv w( →

uv)f(

→

uv) Subject to:

      

• v∈N↓(u) fi(

→

uv) =

v∈N↑(u) fi( →

vu) ∀Ti ∈ A, ∀u fi(

→

TiS) = d ∀Ti ∈ A fi(

→

uv) ≤ f(

→

uv) ≤ c(

→

uv) ∀Ti ∈ A, ∀

→

uv fi(

→

uv), f(

→

uv) ≥ 0 ∀Ti ∈ A, ∀

→

uv

23

Maximum B-B: two-stage linear optimization

Stage 2: compute maximum b-b ratio x:

Maximize x Subject to:

• x|f∗

A| ≤ u∈A yA(u) ≤ |f∗ A|

∀A ⊆ T yA(u) ≤ yB(u) ∀u ∈ B ⊂ A ⊆ T x, yA(u) ≥ 0 ∀u ∈ A ⊆ T

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