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 1

Multicast • Multicast models one-to-many data dissemination in a computer network • Example: live Video Streaming on the Internet 2

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? 4

Min-Cost Multicast with network coding: 0.5 b 0.5 0.5 b b a+b 0.5 0.5 a+b 0.5 a 0.5 0.5 a+b a a 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? 5

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 6

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 of the entire network 7

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 − You are here! – [Li, IEEE INFOCOM 2008] ← • Selfish Links – Induce truthful cost reports from links – Apply the Vickrey-Clarke-Groves Mechanism – Efficient computation of Vickrey prices, ongoing work 8

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 f A – f A has cost | f A | – 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! 9

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] 10

Cross-Monotonicity B A AC AB ABE ABC . . . . . . Travel down any of these Criss-Crossing routes, a node’s cost share should be monotonically decreasing. 11

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 12

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 13

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 14

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 15

The hardness of the problem OPT + C-S + B-B = hard • Direct LP dual is not smooth • Local sharing is not in-core S S 4 4 4 T 1 T 1 T 2 T 2 3 3 16

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 y A ( u ) = | f ∗ u | | A | 17

Directed Networks, negative result • No optimal, cross-monotonic, ( 2 k + ǫ )-budget-balanced √ scheme, ∀ ǫ > 0 e t ∀ ( j 1 , j 2 , ..j l ) ∈ [1 ..h ] l : i t T T 21 T 11 r l1 a p S h c T 12 T 22 T l ... 2 a 1 T lj l e T 1 j 1 n δ ... ... ... δ i s e d T 1 h T lh T 2 h δ ... o n h T 2 j 2 l-partite k : total number of potential multicast receivers k = hl 18

Directed Networks, negative result Probabilistic proof • randomly pick service set A = A 1 ∪ A 2 , show expected B-B factor is low • A 1 : uniformly randomly pick a partite i • A 2 : ∀ j � = i , uniformly randomly pick a user in partite j E A ( � T ij ∈ A y A ( T ij )) = 1 E A ( � T ij ∈ A 1 y A ( T ij )) + E A ( � T ij ∈ A 2 y A ( T ij )) ≤ 2 hE A 2 ,T ij ∈ A 1 ( y A 2 + T ij ( T ij )) + E A 2 ,T ij ∈ A 1 ( � T ij ∈ A 2 y A 2 + T ij ( T ij )) h ( 1 l + δ ) + ( l − 1)( 1 = 3 l + δ ) ( h + l − 1)( 1 2 19 = 4 l + δ ) ≤ √ k

Undirected Networks, Negative Result • No optimal, cross-monotonic, ( 1 2 + ǫ )-budget-balanced scheme, ∀ ǫ > 0 ∀ ( j 1 , j 2 , ..j l ) ∈ [1 ..h ] l : S 1 T lj l T 1 j 1 1 1 1 ... T 2 j 2 20

Undirected Networks, Positive Result k +1 • A (2 k +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 (2 k +1) ζ should be close to 1 k +1 • 2 , almost tight bound • Idea: smooth dual growing, from primal-dual algorithm design 21

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 ) f i ( uv ) = � v ∈ N ↑ ( u ) f i ( vu ) ∀ T i ∈ A, ∀ u   →  f i ( T i S ) = d ∀ T i ∈ A  → → → →  f i ( uv ) ≤ f ( uv ) ≤ c ( uv ) ∀ T i ∈ A, ∀ uv  → → → f i ( uv ) , f ( uv ) ≥ 0 ∀ T i ∈ A, ∀ uv 23

Maximum B-B: two-stage linear optimization Stage 2: compute maximum b-b ratio x : Maximize x Subject to: � x | f ∗ u ∈ A y A ( u ) ≤ | f ∗ A | ≤ � A | ∀ A ⊆ T y A ( u ) ≤ y B ( u ) ∀ u ∈ B ⊂ A ⊆ T x, y A ( u ) ≥ 0 ∀ u ∈ A ⊆ T 24