1. 1 Mechanism Design for Interdomain Routing Rahul Sami Joint work with Joan Feigenbaum, David Karger, Vahab Mirrokni, Christos Papadimitriou, and Scott Shenker.

2. 2 Interdomain Routing Finding routes between autonomous systems (ASes) Currently done with the Border Gateway Protocol (BGP). UMich Qwest Comcast UUNET

3. 3 Comcast Qwest Current Interdomain Routing ASes have pairwise business agreements (customer-provider/peer) Use BGP to: - Learn about their neighbors’ routes - Select a route - Pass on selected route to their neighbors (with modifications) UMich Routing policy implicitly depends on business agreements.

4. 4 Problems with Current Routing Unstable routes (routes keep changing). Inefficient routes. Very difficult to diagnose problems. Uncoordinated, unstructured routing policies can lead to problems: [Griffin et al., Tangmunarunkit et al.,..]

5. 5 High-level Approach Q: Can we systematically include business incentives into interdomain routing? Use tools from: Economics/Game Theory to reason about incentives Theoretical CS to analyze computational efficiency while working within realistic networking constraints. – Prevent problems due to opaque policies. – Enable coordination through multilateral agreements.

6. 6 Economic Mechanism Design Approach to designing systems for self-interested agents. Agent 1 Agent n tntn t1t1 O Private information Outcome

7. 7 Economic Mechanism Design Approach to designing systems for self-interested agents. Strategyproof mechanisms: Regardless of what other agents do, each agent i maximizes her utility by revealing her true private information. Agent 1 Agent n Mechanism p1p1 pnpn tntn t1t1 a1a1 anan O Private information Strategies Payments Outcome

8. 8 Algorithmic Mechanism Design Distributed Algorithmic Mechanism Design (DAMD) [Feigenbaum, Papadimitriou, Shenker ’00] [Nisan & Ronen, ’99] – Polynomial-time computable O( ) and p i ( ) – Computation is distributed across a network. – Need both efficient computation and modest communication requirements. – Centralized model of computation

9. 9 This Work: DAMD for Interdomain Routing Natural real-world problem with distributed control and distributed computation. Real-world constraint: protocol compatibility - BGP is not going to be replaced overnight. - Need to be able to gradually phase in a mechanism. Challenges: Theoretical analysis of protocol compatibility is challenging (but fun!). Models need to capture the essence of routing, yet be simple enough to analyze.

10. 10 Talk Outline Introduction Lowest-Cost Routing - Strategyproof mechanism - BGP-based Computation Model - A BGP-compatible Distributed Algorithm Next-Hop Policy Routing Subjective-Cost Policy Routing Conclusion and Future Directions

11. 11 Model 1: Lowest-Cost Routing ASes private information: Per-packet costs a b source dest. d cost = 2 cost = 3 cost = 4 Goal: Use lowest-cost route for every source-destination pair. How much should transit ASes be paid? [Feigenbaum, Papadimitriou, S., Shenker, PODC ’02.]

12. 12 MD for Lowest-Cost Routing Strategyproofness BGP-based distributed algorithm Lowest-cost paths (LCPs) Per-packet costs {c k }Private Information: Route from every source to every sink.Outputs: Payment to each node { p k }Payments: Objectives: Prior work: Strategyproof mechanism for single source-destination pair. Polynomial-time centralized computation. [Hershberger-Suri ’01]: Faster payment computation for Nisan-Ronen mechanism. [Nisan-Ronen ’01]:

13. 13 Lowest-Cost Routing Mechanism Payments: For each packet from i to j, pay node k on the lowest-cost path a price = c k + Cost of LCP from i to j without using k – Cost of LCP from i to j using k p ij k Under reasonable conditions, there is a unique strategyproof mechanism that always computes LCPs : Inputs: Reported per-packet costs c k Output: Lowest-cost paths (LCPs) k 3 2 8 i b j k 3 2 8 i b j p ij k : Is the price-computation compatible with BGP?

14. 14 Lowest-Cost Routing Mechanism Payments: For each packet from i to j, pay node k on the lowest-cost path a price = c k + Cost of LCP from i to j without using k – Cost of LCP from i to j using k p ij k Under reasonable conditions, there is a unique strategyproof mechanism that always computes LCPs : Inputs: Reported per-packet costs c k Output: Lowest-cost paths (LCPs) k 3 2 8 i b j k 3 2 8 i b j p ij k : Is the price-computation compatible with BGP? YES!

15. 15 BGP-based Computational Model (1) Follow abstract BGP model of [Griffin & Wilfong]: Each AS has a routing table with LCPs to all other nodes: Entire paths are stored, not just next hop. Dest. AS Path AS3AS5AS1 AS7AS2 – Ignore all intradomain routing issues. – Ignore optimizations like route aggregation.

16. 16 BGP-based Computational Model (2) An AS “advertises” its routes to its neighbors in the AS graph, whenever its routing table changes. The computation of a single node is an infinite sequence of stages: Receive routes from neighbors Update routing table Advertise modified routes Complexity measures: − Number of stages required for convergence − Total communication

17. 17 Computing the Prices = c k + Cost of min-cost k-avoiding path from i to j – Cost of LCP from i to j p ij k a b d i k j Key observations: Min-cost k-avoiding path passes through one of i’s neighbor Price can be related to costs and price paid to k by that neighbor, e.g., = + c b + c i p ij k p bj k p ij k Using this, we can show that prices can be computed with local dynamic programming, with nodes exchanging only costs and prices.

18. 18 A BGP-based Algorithm AS3AS5 c(i,1) AS1 c1c1 Dest.Cost LCP and Path Prices LCP cost AS1 Initially, all prices are set to . Then, each node runs the following computation: Final state: Node i has accurate values. p ij k p i1 3 5 Receive routes and prices from neighbors Update routes and prices Advertise modified routes and prices

19. 19 Performance of Algorithm d’ = maximum length of min-cost k-avoiding path d = maximum length of LCP Our algorithm computes the VCG prices correctly, and converges in max(d,d’) stages. Theorem : Measuring a recent Internet AS graph, we found d=9, d’=11 (assuming uniform costs). Conclusion: Mechanism is BGP-compatible!

20. 20 Strategies in Computation Mechanism is strategyproof : ASes have no incentive to lie about c k ’s. However, payments are computed by the strategic agents themselves  they may be able to manipulate the mechanism in other ways. One approach [Mitchell, S., Talwar, Teague]: Digitally sign every message. A better solution [Shneidman & Parkes, 2004]: Prevent manipulation using digital signatures and redundant communication.

21. 21 Talk Outline Introduction Lowest-Cost Routing Next-Hop Policy Routing - Strategyproof Mechanism - Incompatibility with BGP Subjective-Cost Policy Routing Conclusion and Future Directions

22. 22 Model 2: Next-Hop Policies cust. peer i j Lowest-cost routing is inadequate Routing policies depend on other factors, e.g., customer-provider/peer relationships Need to express: “AS i prefers any route through customer over any route through peer.” [Feigenbaum, S., Shenker, PODC ’04.]

23. 23 Next-Hop Policy Routing Next-hop preference model: Each AS i assigns a value u i (P ij ) to each potential route P ij. u i (P ij ) depends only on the first hop of path P ij. Captures preferences due to customer-provider/peer agreements. cust. peer i j 5 2

24. 24 Next-Hop Policies: Optimal Tree  Maximize total welfare W =  u i (P ij ).  For each destination j, {P ij } forms a tree. Mechanism goals:

25. 25 Next-Hop Policies: Optimal Tree  Maximize total welfare W =  u i (P ij ).  For each destination j, {P ij } forms a tree. Mechanism goals: i b a j c 5 8 3 3 Optimal tree: Maximum-weight directed spanning tree (MDST) 6 2 5 1

26. 26 Strategyproof Mechanism T -k = MDST in G – {k}where Under reasonable conditions, there is a unique strategyproof mechanism that maximizes welfare: Inputs: Reported valuations u i (.)  weighted graph G Routes: Maximum-weight directed spanning tree (MDST) T* in G Payments: Node k is paid an amount p k = W(T*) – u k ( T*) – W(T -k ) Is there a BGP-based algorithm to compute prices?

27. 27 Proving Hardness for BGP-based Routing Algorithms Requirements for any BGP-compatible algorithm: [P1] Each AS uses O(l) space for a route of length l. [P2] The routes converge in O(log n) stages. [P3] Most changes should not have to be broadcast to most nodes: i.e., few nodes can individually trigger  (n) updates. For “Internet-like” graphs:  O(1) average degree  O(log n) diameter Small numeric values, and remains hard if perturbed slightly. Hardness results must hold:

28. 28 Dynamic Sensitivity B3B3 R3R3 9 9 B4B4 R4R4 9 9 7 7 9 9 9 9 7 j B1B1 R1R1 B2B2 R2R2 5 3 5 2 Blue spanning tree is MDST. Red spanning tree is MDST if any B x is removed. Change in any edge changes the price of every B x.  Any change triggers update messages to all blue nodes! Theorem: A distributed algorithm for the MDST mechanism cannot satisfy property [P3]. Conclusion: MDST mechanism not BGP-compatible.

29. 29 Network Construction (1) (a) Construct 1-cluster with two nodes: B R 1-cluster red port blue port (b) Recursively construct (k+1)-clusters: blue port red port L-1 BRBR k-cluster (k+1)-cluster L- 2k -1 L  2 log n + 4

30. 30 Network Construction (2) (c) Top level: m-cluster with n = 2 m + 1 nodes. BR m-cluster L- 2m -1 L- 2m -2 j Destination Final network (m = 3): red port blue port B R 9 9 B R 9 9 7 7 9 9 9 9 7 j B R B R 5 3 5 2

31. 31 Optimal Spanning Trees Lemma: W(blue tree) = W(red tree) + 1  W(any other sp.tree) + 2 Proof: If a directed spanning tree has red and blue edges, we can increase its weight by at least 2: B B R k-cluster (k+1)-cluster L- 2k -3 B k-cluster (k+1)-cluster L- 2k -1 L- 2k -3

32. 32 Proof of Theorem B R 9 9 B R 9 9 7 7 9 9 9 9 7 j B R B R 5 3 5 2 MDST T* is the blue spanning tree. For any blue node B, T -B is the red spanning tree on N – {B}. A small change in any edge, red or blue, changes  Any change triggers update messages to all blue nodes! p B = W(T*) – u B (T*) – W(T -B )

33. 33 Talk Outline Introduction Lowest-Cost Routing Next-Hop Policy Routing Subjective-Cost Policy Routing - A “payment-less” Mechanism Conclusion and Future Directions

34. 34 Model 3: Subjective Costs Each AS i assigns a subjective cost c i (k) to every transit AS k. Mechanism design goals: – Minimize total cost C =  i c i (P ij ). – Routes to destination j form a tree T. Minimizing C is computationally hard even for very restricted cases: costs in {0,1} or costs in {1,2}. Special case: Forbidden-set routing If i uses route P ij to destination j, it incurs a cost c i (P ij ) =  k  Pij c i (k). [Feigenbaum, Karger, Mirrokni, S., WINE’05.]

35. 35 Convex Subjective Costs Special case: Weighted average of two objective metrics (e.g., latency and loss rate): – Each transit AS k has two “lengths” l 1 (k), l 2 (k). – c i (k) = λ i l 1 (k) + (1 – λ i )l 2 (k) – λ i  [0,1] is privately known to i.  Hard to find optimal tree or (1+ε)-approximation. l 1 only 50% l 1 50% l 2 l 1 =1, l 2 =10 l 1 =10, l 2 =1 a b d j l 1 =2, l 2 =2 e f l 2 only

36. 36 A Few Trees are Sufficient Allow r > 1 trees to each destination j. O(log n) trees sufficient to include a (1+ε)-approximately optimal route for every i. Theorem: Relax the problem definition: Each AS i has r possible routes to j, and uses the one which it likes best. – Each tree is LCP-tree for some cost metric  easy BGP-based implementation. – Tree construction does not use private information  trivially strategyproof ! This is a feasible mechanism for interdomain routing!

37. 37 Talk Outline Introduction Lowest-cost Routing Next-Hop Policy Routing Subjective-Cost Policy Routing Conclusion and Future Directions

38. 38 Summary An approach combining incentive analysis, distributed algorithmics, and practical constraints is feasible and useful to understand and improve the interdomain routing system. Explores trade-off between expressiveness and routing complexity. “Practical” issues like protocol compatibility can be modeled and analyzed.

39. 39 Related Work on Incentives in Interdomain Routing Policy routing convergence/oscillation [Griffin, Gao, Rexford, Shepherd, Wilfong,...] Convergence with next-hop policies [Feamster, Johari, Balakrishnan] Repeated-game Models of Routing [Afergan, Wroclawski] Strategies in Computation [Shneidman, Parkes]

40. 40 Some Open Questions Is there a tractable, sufficiently expressive class of policies? Can the MDST be approximated? Can we take into account both source preferences and transit costs? Study alternatives to strategyproof mechanisms Does routing complexity generally increase with subjectivity?