CMSC 858F: Algorithmic Game Theory Fall 2010 BGP and Interdomain Routing

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1 CMSC 858F: Algorithmic Game Theory Fall 2010 BGP an Interomain Routing Instructor: Mohamma T. Hajiaghayi Scribe: Yuk Hei Chan November 3, Overview In this lecture, we cover BGP (Borer Gateway Protocol) an interomain routing, an iscuss their relation to game theory. 2 Introuction The Internet can be viewe as a collection of clous, where each clou is an autonomous system (AS), a network (or a set of networks) that is uner a single aministration. For example, the whole UMD network can be an AS, while for larger corporations like AT&T, the whole corporate network can be ivie into several ASes, say AT&T California, AT&T Texas, an so on. Each AS has an officially registere AS number (ASN). Currently (as of 2010) there are over 35,000 registere ASNs in use. IGP (Interior Gateway Protocol) is use insie one AS. It is use to figure out how o to route within an AS. IGRP (Interior Gateway Routing Protocol) an OSPF (Open Shortest Path First) are use as an IGP. They are istancebase policies. IGRP, together with its enhance version, EIGRP (Enhance IGRP), are Bellman-For base protocols. They are istance-vector routing protocols. In each router, there is a routing table which is a vector of size n (n is the number of routers in the AS). This vector inicates the istance to other routers in the AS, as well as the next hop neighbor to sen the packet to. The router oes not possess the full information about the network topology. The istance-vector is avertise to its neighbors to perform upates to the routing table. IGRP an EIGRP are supporte by Cisco routers. OSPF, on the other han, is a link-state routing protocol. It saves the current state (ajacency matrix) of the whole worl (which is the whole AS). 1

2 It is a Dijkstra base protocol an is more ominant. The ajacency matrix is upate with the istance / capacity of the router s links. It is more global compare to EIGRP. If there are several shortest paths, it ivies the payloa into parts an utilize all shortest paths so it is faster. OSPF is also supporte in Cisco routers. In upating the istance metrics, we assume metricity (or triangle inequality), i.e. (a, c) (a, b) + (b, c) for all router a, b an c in the AS. BGP (Borer Gateway Protocol) is the sole protocol use for routing between omains. It replace EGP (Exterior Gateway Protocol) since 1994 an BGP version 4 is now the accepte stanar. It allows each AS to efine its own preference for routing policy. The policy oes not nee to ahere to a istance-base policy like the ones in Interior Gateway Protocol (IGP) metrics an Exterior Gateway Protocol (EGP). In BGP, each AS becomes a noe. By making sure the AS number oes not appear in a path more than once, loops are prevente. The analysis of the robustness of policy choices in BGP has implication towars the overall efficiency an the functionality of the Internet. Also, stability correspon to Nash equilibria an that s why we stuy BGP. 3 Theoretical Moel We moel the whole system as an unirecte graph on ASes, where each AS is a noe an we want to fin some route to the estination. The goal is to esign mechanism so as to have goo behavior. Consier the BGP routing mechanism where is the estination: 1. avertise itself 2. For all router v : Iteratively receives upates about path to Receives status upates Choose the best path an upate the forwaring table (accoring to some policy) Announce the best path to its neighbors similar to IGRP As a simple example, noe i (i = 1, 2, 3) prefers the route i(i + 1) to i to the estination : Let s say at the beginning, 3 picks the route 3 (as there are currently no paths chosen). Then, 2, on seeing the choice of 3, picks the route 23 (since 23 is preferre by 2 to the route 2). 1 picks the route 1 (Figure 3, left). Then 3, on seeing the choice of 1, changes its min an picks the route 31. Next, seeing that the route 23 is gone, 2 has to pick the route 2 (Figure 3, right). In turn, 1 sees the opportunity to change is route to 12, which forces 3 to change its choice. This switching among route on each router s preference list goes on forever which means the system is not stable. 2

3 3 31 Preference 3 3 list of routes Figure 1: Not all choices of preference list lea to stable paths. 4 The Stable Path Problem (SPP) The stable path problem (SPP) is as follows: Input: an unirecte graph G = (V, E). For each v V, P v is the set of permitte (simple) paths from v to the estination vertex. For each v V, there is a ranking function λ v efine over P v. If λ v (P 1 ) < λ v (P 2 ) then P 2 is a more preferre permitte path than P 1. Empty path ɛ P v is permitte an ranke lowest: λ v (ɛ) = 0, while λ v (P) > 0 for P ɛ. P 1, P 2 P v = λ v (P 1 ) λ v (P 2 ). (each path receives a istinct ranking) If the en point of a path P is the same as the start point of a path Q, the concatenation of the two paths is enote by PQ. If P = (1, 2) an Q = (2, 3) then PQ = (1, 2, 3). Note that ɛp = Pɛ = P. A path assignment is a function Π that maps each noe u V to a path Π(u) P u (with the special case Π() = ()). Initially Π(u) = ɛ which means u is not assigne a path to the estination. The set of paths choices(π, u) for u = is {()} an {(u, v)π(v) {u, v} E} P u otherwise. This represents all possible permitte paths at u that can be forme by extening the paths assigne to neighbors of u. Given a noe u, suppose W is a subset of the permitte paths P u such that each path in W has a istinct next hop. Then the best path in W is efine to be best(w, u) = P W with maximal λ u (P) for W an ɛ otherwise. The path assignment Π is stable at noe u if Π(u) = best(choices(π, u), u). Note that if Π is stable at noe u an Π(u) = ɛ, then the set of choices at u must be empty. The path assignment Π is stable if it is stable at each noe u. Essentially, Π = (P 1,..., P n ) where Π(u) = P u (assume 0 is the estination). If Π is stable an Π(u) = (u, w)p, then Π(w) = P. Hence, any stable assignment efines a tree roote at the estination (each noe has one outgoing ege that points towars the estination), although it is not always the case that it is a shortest path tree. 3

4 u n 1 R n 1 Q n 1 u 0 Q 0 R 0 u 1 0 Q 1 R 1 0 Q 2 u 2 Figure 2: (Left) A isagree gaget; (Right) A ispute wheel. The stable paths problem S = (G, P = v P v, λ) is solvable if there is a stable path assignment. The example in Figure 3 oes not have a stable path assignment. Theorem 1 (Griffin, Shepher, Wilfong, TON 2002) SPP is NP-Complete (prove by reucing it to 3-SAT). When oes SPP amit a unique solution? Let s look at a configuration with more than one solution. A isagree gaget is shown in Figure 2, left. In this configuration, there are 2 stable paths, namely (2, 1, 0) an (1, 2, 0). The isagree gaget can be generalize into a ispute wheel, as shown in Figure 2, right: 1. A rim path R i is a path from u i to u i+1 2. Spoke path Q i P u i 3. R i Q i+1 P u i 4. λ u i (Q i ) < λ u i (R i Q i+1 ) Theorem 2 If the stable path problem S has no ispute wheel configuration, then there is a unique solution. Note that if S has no ispute wheel, then it is solvable. Roughly speaking, having no ispute wheel implies safety an robustness. There are two types of relationship between entities on the Internet, namely provier-customer an peer-to-peer (Figure 3). Gao-Rexfor conition is a reasonable assumption in the context of connectivity of the Internet, which states if there is no customer-provier (irecte) cycle, then there is no ispute wheel configuration, which implies a stable assignment. We can achieve this by filtering of paths, e.g. not avertising your peers to your provier. We can efine a game such that Nash equilibria of the game are precisely the stable solutions in the equivalent SPP formulation. 4

5 Peers Proviers Peers Customers Figure 3: Provier-customer relationship (soli line) an peer-to-peer relationship (ashe line). V i P 2 u 2 u u 1 D i Figure 4: A consistent path. 5 Greey Algorithm in the Absence of Dispute Wheel In the absence of ispute wheel, a greey algorithm that is base on the iea of expaning the tree can be use to fin the solution to the SPP. Starting with V 0, which contains only the estination 0, we construct larger an larger set V i, such that {0} = V 0 V 1 V k, an for v V i, Π(v) V i (i.e. the whole path stays insie V i ). If u V V i an P P u, then P is sai to be consistent with current (partial) Π i if P = P 1 (u 1, u 2 )P 2, where P 1 is a path in the igraph inuce by V V i, u 2 V i, u 1 V V i, {u 1, u 2 } E an P 2 Π(u 2 ) (Figure 4). Call a path P a irect path to V i if P 1 = (in this case u = u 1 ). Let D i be the set of noes with irect path to V i. Assuming every noe has a non-empty permitte path to the origin, D i is non-empty. Let H i D i be the set of vertices u such that a irect path of u is preferre among all consistent paths of u. For the greey algorithm to work, we want H i to be non-empty at each iteration. What if H i = but D i? Since u 0 D i H i, there is a vertex u 1 V V i an path R 0 V V i from u 0 to u 1, such that {u 1, v 1 } E, v 1 V i an P i (v 1 ) Π(v 1 ). Let the path from u 0 to a vertex in V i an on to the estination be Q 0. Since Q 0 is not the preferre path, we have λ(r 0 Q 1 ) > λ(q 0 ), where R 0 is a path in V V i that leas from u 0 to u 1, an Q 1 is a path from u 1 to v 1 an on to the estination (Figure 5). Then consier u 1. Again since 5

6 Q 0 v 0 u 0 Q 1 v 1 R 0 V i u 1 Figure 5: The first step to a ispute wheel. u 1 D i H i, there exists u 2 V V i, R 1 V V i from u 1 to u 2, Q 2, v 2 V i with {u 2, v 2 } E an λ(r 1 Q 2 ) > λ(q 1 ). The argument continues an eventually forms a ispute wheel (although u 0 may not involve in the ispute wheel). This shows that H i an proves that the greey algorithm works in the absence of ispute wheel. References [1] Timothy G. Griffin, F. Bruce Shepher, Goron Wilfong. The stable paths problem an interomain routing. IEEE/ACM Transactions on Networking (TON), Vol. 10(2), [2] L. Gao, J. Rexfor. Stable Internet Routing Without Global Coorination. ACM SIGMETRICS, June