1 A BGP-based Mechanism for Lowest-Cost Routing Rahul Sami Yale University Joint work with: Joan Feigenbaum Yale Christos Papadimitriou U. C. Berkeley Scott Shenker ICSI

2 Lowest-Cost Routing Mechanism-design Problem Inputs: Transit costs Outputs: Routes, Payments Qwest Sprint UUNET WorldNet Agents: Transit ASs

3 Strategyproof Mechanism Design Agent i chooses strategy a i to maximize her utility. Strategyproof mechanisms: Regardless of what other agents do, each agent i maximizes her utility by revealing her true private type information t i. Agent 1 Agent n Mechanism p 1p 1 p np n t nt n t 1t 1 a 1a 1 a na n O Private “types” Strategies Payments Output

4 Algorithmic Mechanism Design Polynomial-time computable O( ) and p i ( )  Introduced computational efficiency into mechanism-design framework. Centralized model of computation AMD (Nisan-Ronen ’99): Distributed AMD (Feigenbaum-Papadimitriou-Shenker ’00) Computation is distributed across the network “Good network complexity”: − Polynomial-time local computation − Modest in total number of messages, messages per link, and maximum message size.

5 Outline VCG Mechanism for Lowest-Cost Routing “BGP-based” Computational Model DAM for Lowest-Cost Routing Open Questions

6 Problem Statement Strategyproofness “BGP-based” distributed algorithm Lowest-cost paths (LCPs) Per-packet costs {c k } Agents’ types: {route(i, j)}Outputs: (Unknown) global parameter: Traffic matrix [T ij ] {pk}{pk} Payments: Objectives:

7 Previous Work Nisan-Ronen, 1999 Single (source, destination) pair Links are the strategic agents “Private type” of l is c l (Centralized) strategyproof, polynomial-time mechanism Hershberger-Suri, 2001 p l  d G|c l =  - d G|c l =0 Compute m payments as quickly as 1

8 Our Formulation vs. NR, HS Nodes, not links, are the strategic agents. All (source, destination) pairs Distributed “BGP-based” algorithm More realistic model Advantages: Deployable via small changes to BGP

9 Notation LCPs described by indicator function: 1 if k is on the LCP from i to j, when cost vector is c 0 otherwise c Ι   (c 1, c 2, …c k-1, , c k+1 …, c n ) { I k (c; i,j)  k

10 A Unique VCG Mechanism For a biconnected network, if LCP routes are always chosen, there is a unique strategyproof mechanism that gives no payment to nodes that carry no transit traffic. The payments are of the form p k =  T ij, where = c k I k (c; i, j) + [  I r (c Ι  ; i, j ) c r -  I r (c; i, j ) c r ] Theorem 1: p ij k rr i,j p ij k k

11 Features of this Mechanism Payments have a very simple dependence on traffic [T ij ]: payment p k is the sum of per-packet prices. Cost c k is independent of i and j, but price depends on i and j. Price is 0 if k is not on LCP between i, j. Price is determined by cost of min-cost path from i to j not passing through k (min-cost “k-avoiding” path). p ij k k k k

12 BGP-based Computational Model (1) Follow abstract BGP model of Griffin and Wilfong: Network is a graph with nodes corresponding to ASes and bidirectional links; intradomain-routing issues are ignored. Each AS has a routing table with LCPs to all other nodes: Entire paths are stored, not just next hop. Dest. LCP LCP cost AS3AS53AS1 AS7AS22

13 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

14 Constructing k-avoiding Paths Three possible cases for P -k (c; i, j): j a b d i k i ’s neighbor on the path is (a) parent (b) child (d) unrelated In each case, a relation to neighbor’s LCP or price, e.g., (b) = + c b + c i is the minimum of these values. p ij k p bj k p ij k = c k + Cost ( P -k (c; i,j) ) – Cost ( P(c; i,j) ) p ij k in tree of LCPs to j.

15 A “BGP-based” Algorithm AS3AS5 c(i,1) AS1 c1c1 Dest.cost LCP and path prices LCP cost AS1 LCPs are computed and advertised to neighbors. Initially, all prices are set to . Each node repeats: − Receive LCP costs and path prices from neighbors. − Recompute path prices, selecting lowest prices. − Advertise changed prices to neighbors. Final state: Node i has accurate values. p ij k p i1 3 5

16 Performance of Algorithm d’ = max i,j,k || P -k ( c; i, j ) || d = max i,j || P ( c; i, j ) || Our algorithm computes the VCG prices correctly, uses routing tables of size O(nd) (a constant factor increase over BGP), and converges in at most (d + d’) stages (worst-case additive penalty of d’ stages over the BGP convergence time). Theorem 2:

17 Open Question: Strategy in Computation Mechanism is strategyproof : ASes have no incentive to lie about c k ’s. However, payments are computed by the strategic agents themselves. How do we reconcile the strategic model with the computational model? First approach : Digital Signatures [Mitchell, Sami, Talwar, Teague] Is there a way to do this without a PKI?

18 Open Question: Overcharging In the worst case, path price can be arbitrarily higher than path cost [Archer&Tardos, 2002]. This is a general problem with VCG mechanisms. Statistics from a real AS graph, with unit costs: Mean node price : 1.44 Maximum node price: 9 90% of prices were 1 or 2 How do VCG prices interact with AS-graph formation? Overcharging is not a major problem!