cooperative interdomain traffic engineering using nash bargaining and decomposition

IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 341

Cooperative Interdomain Traffic Engineering UsingNash Bargaining and Decomposition

Gireesh Shrimali, Aditya Akella, Member, IEEE, and Almir Mutapcic, Member, IEEE

Abstract—We present a novel approach to interdomain trafficengineering based on the concepts of Nash bargaining and dualdecomposition. Under this scheme, ISPs use an iterative procedureto jointly optimize a social cost function, referred to as the Nashproduct. We show that the global optimization problem can beseparated into subproblems by introducing appropriate shadowprices on the interdomain flows. These subproblems can then besolved independently and in a decentralized manner by the indi-vidual ISPs. Our approach does not require the ISPs to share anysensitive internal information, such as network topology or linkweights. More importantly, our approach is provably Pareto-effi-cient and fair. Therefore, we believe that our approach is highlyamenable to adoption by ISPs when compared to past approaches.We also conduct simulation studies of our approach over severalreal ISP topologies. Our evaluation shows that the approachconverges quickly, offers equitable performance improvementsto ISPs, is significantly better than unilateral approaches (e.g.,hot-potato routing) and offers the same performance as a central-ized solution with full knowledge.

Index Terms—Cooperative game theory, dual decomposition,hot-potato routing, interdomain traffic engineering (TE), ISPpeering, Nash bargaining, Nash equilibrium.

I. INTRODUCTION

A KEY component of operating and managing any ISP net-work is the ability to control how traffic enters or leaves

the network. This is critical to ensuring that the ISP can offergood performance and reliability even in the face of internal orexternal failures and overload.

BGP provides networks with a limited set of mechanismsto achieve this control (e.g., local prefs for outbound control,MEDs and AS path prepending for inbound control). However,these mechanisms only offer ISPs unilateral control over traffic.Unfortunately, unilateral decisions of neighboring networksmay have undesirable interactions, and may result in unstable

Manuscript received March 26, 2008; revised February 25, 2009; approvedby IEEE/ACM TRANSACTIONS ON NETWORKING Editor J. Yates. First publishedOctober 30, 2009; current version published April 16, 2010.

G. Shrimali is with the Indian School of Business, Hyderabad 500032, India(e-mail: [email protected]).

A. Akella is with the Department of Computer Science, University of Wis-consin, Madison, WI 53706 USA (e-mail: [email protected]).

A. Mutapcic is with the Department of Electrical Engineering, Stanford Uni-versity, Stanford, CA 94305 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNET.2009.2026748

routing [1], [2], poor performance [3], and huge, unpredictableshifts in network traffic volumes [4].1

Recently, it has been argued that supporting dynamic con-trol over interdomain traffic in a stable, efficient and predictablemanner requires a new interdomain traffic engineering architec-ture that is based on explicit coordination between ISPs [2], [5],[6]. In this approach, neighboring ISPs exchange informationabout interdomain traffic volumes and routes, and participate ina simple “negotiation protocol” to arrive at mutually acceptableroutes for the traffic between them (see Section II for more de-tails). It has been shown that such explicit coordination can si-multaneously help both networks [2], [6].

These seminal studies establish the potential benefits of co-ordinated interdomain traffic engineering (TE). Unfortunately,realizing cooperation among ISPs in practice is not straightfor-ward since ISPs also compete against each other, and their com-petitive concerns must be explicitly accounted for. In particular,we note that cooperative interdomain TE approaches must sat-isfy the following criteria for ISPs to adopt them.2

A. Minimum Information Revealed

ISPs regard their network structure, link capacities, and linkweights as “sensitive internal information,” crucial to main-taining their competitive edge. Therefore, ISPs must be able toperform cooperative TE without having to reveal their sensitiveinformation.

B. Efficiency

Cooperative approaches must ideally result in Pareto-efficientoperating points. By this, we mean that the resulting allocationof traffic across interdomain routes must lie on the boundaryof the feasible outcomes—on this boundary, we cannot makeone ISP better off without disadvantaging the other. Pareto-effi-ciency ensures that network resources are used in the most effi-cient manner by both ISPs. Note that efficient network usage isalso the driving goal of intradomain traffic engineering.

C. Fairness

Any interdomain TE approach should yield an operatingpoint that is provably fair. By fair, we mean that cooperationshould yield equitable performance gains to the participantswhen compared to their default TE strategies (e.g., both ISPs

1Many peering negotiations have broken down over the issue of who willlisten to MEDs. We encourage the interested reader to consult the examples in[2] to get a better feel for various issues arising in unilateral interdomain trafficengineering.

2Unless otherwise specified, our focus in this paper is on a pair of neighboringISPs.

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342 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010

employing unilateral control). Approaches that yield dispro-portionate benefits are likely to be spurned by the ISP that getsless benefit out of the arrangement.

To the best of our knowledge, no single approach for interdo-main traffic engineering can provide information hiding alongwith fairness and efficiency (i.e., criteria 1–3 listed above). Ex-isting approaches [2], [6] at best satisfy the first criteria, but notthe other two. In this paper, we present a new cooperative in-terdomain TE approach that can provably offer these three de-sirable properties. Therefore, we believe that our approach ishighly amenable to adoption by ISPs.

Our approach uses ideas from multicriteria optimization [7]and axiomatic bargaining [8]. Like past studies, we assume thatISPs can improve their local performance by bargaining (or ne-gotiating) about the traffic flow distribution on their peeringlinks. Our first insight is that we can use the well-known conceptof Nash bargaining [9], [10] to do so. Under this scheme, theISPs agree to jointly optimize a social cost function (defined asany function of both the utilities), known as the Nash product,which is essentially the product of the utility functions of thetwo ISPs. The key advantage of using this approach is that thesolution is guaranteed to provide Pareto efficiency and fairness.When the ISPs’ utilities (measures of performance, such as av-erage delay or maximum load on a link) are directly compa-rable, this solution is min–max fair, i.e., the gains from coop-eration are equal. However, when the utilities are not compa-rable, it still provides a Pareto-efficient solution that is propor-tionally fair [11]. By this, we mean that the gains from coopera-tion to individual ISPs are equal after some (automatic) suitablescaling of the utilities. This scaling is endogenous to the solutionand, therefore, is highly desirable. We emphasize that though theNash bargaining solution results from optimizing a social costfunction, it is very different from the globally optimum solution,sometimes referred to as the social optimum, which would beobtained if both the networks were controlled by the same ISP.The globally optimum solution, though Pareto-efficient, can bevery unfair [2].

This leaves us with the issue of not revealing critical internalinformation. Our insight here is that we can use dual decomposi-tion [12] to transform the joint optimization of the Nash productinto a procedure with precisely this property, as follows. Theglobal optimization problem can be decomposed into two inde-pendent subproblems by recognizing the coupling flows (theseare the flows crossing between ISP domains) and introducingappropriate Lagrange multipliers (or shadow prices) [12]. Thesesubproblems can now be assigned to the ISPs to be solved in adecentralized manner. These have the critical feature that theyare completely local—an ISP’s assigned subproblem dependsonly on its own network—thus, the ISPs don’t have to share crit-ical internal information. Relying on this insight, we develop aniterative procedure based on the subgradient method [13] where,given Lagrange multipliers, the ISPs independently optimizetheir local subproblems to come up with their required couplingflows. The Lagrange multipliers are then updated using the sub-gradient method, which uses the difference in the two sets ofrequired flows to determine the magnitude of the update. Theupdate can be done in a decentralized manner. After the update,the ISPs again try to optimize their local subproblems. We show

that this process converges in finite time to a fair and Pareto-ef-ficient allocation.

We evaluate the effectiveness of our approach using simu-lations over realistic ISP topologies. Through simulations, weshow that our approach significantly outperforms unilateral ap-proaches such as the commonly used hot-potato or shortest pathrouting (where ISPs route to nearest peering location in termsof link weights) as well as the Nash equilibrium setting (whereISPs optimize local objectives while playing best responses toeach other [8]). For the case where ISPs employ similar utilities,we compare our solution against centralized optimal routing(where a central arbitrator optimizes the common objectiveacross both ISP networks). We also confirm the proportionalfairness guarantees of our solution via simulation experiments.

We note that another desirable property is that of incentivecompatibility, which means that the participating ISPs haveno incentive to lie or cheat. Without this guarantee, ISPs can“game” any interdomain TE protocol to gain unfair advantage.It is a well-known fact that achieving fairness, efficiency andincentive compatibility together is impossible [14]. However,it is possible to achieve two out of these three criteria. As afirst step, in this paper, we focus on fairness and efficiency. Weassume that ISPs are willing to cooperate with each other, andwill not resort to lying, as long as cooperation can yield betterperformance than the default uncooperative mode of operation.However, we recognize the importance of incorporating incen-tive compatibility into the approach and leave it for future work(further details in Section V).

The organization of the paper is as follows. Section II ex-amines related work. Section III presents the Nash bargainingand dual decomposition framework. Though this paper is fo-cused on a new traffic engineering approach, we suggest someguidelines for implementation in Section IV, including deploy-ment and computation/communication complexity. Similarly,though our paper shows the existence of a new traffic engi-neering approach, Section V looks at the extensions of our basicapproach, including multiple ISP topologies and error handling.Section VI then presents numerical results for our basic ap-proach. The paper concludes with Section VII.

II. RELATED WORK

A. Interdomain TE: A Single ISP’s View Point

Several papers on Interdomain traffic engineering have fo-cused on studying the problem from the point of view of oneof the participants (see, for example, [15]–[17]). These papersaddress issues such as tweaking OSPF weights to achieve fine-grained control over egress points for interdomain traffic [15],AS-path prepending to control the ingress points of interdomaintraffic [17], and best common practices for achieving predictableand stable route selection for interdomain traffic [16]. These pa-pers differ from our paper in the key aspect that we focus on thebenefits of bilateral cooperation among ISPs, while the abovepapers focus on tweaking the unilateral decisions of a singleISP.3 We do note that our technique can operate in conjunction

3Note that these kind of approaches can not only lead to suboptimality, butalso to long-term oscillations due to ISPs compensating for TE changes madeindependently in other ISPs [18]–[20].

SHRIMALI et al.: COOPERATIVE INTERDOMAIN TRAFFIC ENGINEERING 343

with the above approaches: once our technique determines thetraffic volumes to route via different exit points, the above ap-proaches can be used to tune the configurations of routers inorder to achieve the desired effect.

B. Interdomain TE Based on Cooperation

The paper that is perhaps the closest in its goal to our workis Mahajan et al.’s “negotiation-based routing.” In [2] and [6],Mahajan et al. propose an approach where peering ISPs use a“negotiation protocol” to exchange opaque preference classesfor interdomain flows. Using these opaque preference classes,an ISP can indicate the preferred entry points for traffic arrivingfrom its neighbor. No other internal information is exposed.The negotiation protocol proceeds in iterations, with ISPs takingturns in stating their preference for each interdomain flow, untilthey arrive at mutually acceptable mappings of all interdomainflows to network entry points. Thus, cooperative traffic engi-neering is achieved.

This approach was shown to work well in practical set-tings. However, it suffers from the following limitations: It isheuristic-based and, so, does not offer any provable guaran-tees. First, it does not guarantee that the mutually acceptableoutcome lies on the Pareto frontier. Second, it does not makethe idea of fairness concrete. For example, if the ISPs are op-timizing directly comparable objective functions then the finaloutcome should satisfy the well-known min–max criteria whichguarantees equal gains from cooperation. Fairness becomeseven harder to provide when the ISPs are optimizing differentobjective functions. Our work directly addresses the aboveissues.

C. Optimization-Based TE Approaches

A few research studies have explored the applicability of opti-mization techniques to traffic engineering problems. Represen-tative examples include [21]–[23]. The focus of these papers ison intradomain traffic engineering. Reference [21] shows howto cast network-wide traffic engineering goals as optimizationproblems, and how to transform the results into OSPF weights.References [22] and [23] show how to jointly optimize mul-tiple objectives in traffic engineering (such as congestion con-trol and routing, or pricing and routing). Our paper extends thisbody of work in a new direction: joint-optimization of trafficengineering objectives of multiple ISPs. Our contribution is inshowing that this optimization is separable, and therefore, canbe performed in a distributed fashion without requiring the par-ticipants to reveal any sensitive internal information.

D. Incentive Compatibility

As we have mentioned, incentive compatibility is an impor-tant issue, yet to be addressed by us. Prior literature [24], [25]has addressed this issue in relation to interdomain traffic engi-neering. However, this work does not look at the issues of effi-ciency and fairness. Further, it reduces every AS to a single link,and it is not clear how it would scale to real topologies. Finally,this work leaves the implementation of the resulting protocol toindividual ISPs, and it is not clear how strategy-proof that is.

Fig. 1. The model.

E. Nash Bargaining in Other Applications

The application of Nash bargaining to multicriteria optimiza-tion is not new. It has been applied to many problems in net-working. Reference [26] applied it to ensure fairness in a net-work flow control problem. Reference [27] applied it to allo-cate bandwidth fairly. It has also been shown in the influentialwork of Kelly [11] that Nash bargaining ensures proportionalfairness in a TCP setting. To the best of our knowledge, oursis the first work to apply Nash bargaining to interdomain trafficengineering.

F. Price of Anarchy

Recently, there has been a lot of interest in price of anarchyresults where the efficiency loss, defined as the normalized dis-tance of utilities under Nash equilibrium from global optimum(routing realized if both the networks were controlled by thesame ISP), is bounded [28] as well as measured [29] for a singleunified network. Given our focus on improving the utilities ofall involved ISPs, we feel that comparisons to global optimumare somewhat of theoretical interest only and, in order to keepour work focused, have not pursued them.

III. INTERDOMAIN TRAFFIC ENGINEERING USING

NASH BARGAINING AND DECOMPOSITION

A. The Model

We model the interaction between two ISPs: andas shown in Fig. 1. These ISPs are optimizing utilities and ,respectively. As mentioned in Section I, these utilities are func-tions of network flows and are related to some measure of net-work performance. These utilities could mean different things tothe ISPs. For example, for one ISP the utility could be related tothe average delay in the network, and for the other ISP the utilitycould be related to the maximum link load in the network. TheISPs optimize these utilities under the flow conservation con-straints, i.e., flows from all sources to all destinations must berouted. To simplify exposition, in the following description, weassume that the ISPs employ MPLS-like routing (see Section IVfor a discussion on how to employ our approach in conjunctionwith OSPF).

We make the common assumption that the utilities are eitherconvex or concave functions of link flows (and, hence, of in-terdomain flows by virtue of nested optimization [13]) and thatthe ISPs are, respectively, minimizing or maximizing these util-ities. Since minimizing a convex function is equivalent to max-imizing the negative of the function, which is concave, the two


notions are interchangeable. We note that the commonly usedmetrics of maximum link load in the network and the averagenetwork delay (using convex link per unit delay functions, e.g.,the per-unit delay in an queue) are examples of convexutilities.

We assume that needs to send flows to on aper-destination basis: is a vector of flows to be sent to eachof the destinations in from .4 Similarly, needsto send flows to on a per-destination basis: is avector of flows to be sent to each of the destinations in from

. Even though the ISPs may have multiple peering links,to facilitate easier understanding, we explain our model usingtwo bidirectional peering links (Fig. 1). The model generalizesreadily to multiple peering links. We assume that splits

so that goes on the upper link and goes onthe lower link. Similarly splits so that goes on theupper link and goes on the lower link.

Optimizing for the utilities would be trivial if the ISPs hadno interaction. Then, they could optimize their respective util-ities independently. What makes it complicated is the interac-tion of the ISPs through flows sent between each other. Theseflows make the ISPs utilities interdependent. These flows be-tween the ISPs are sometimes referred to as coupling flows sincethey cause the ISPs optimization problems to be coupled.

If the ISPs are myopic, i.e., they employ unilateral approachestoward interdomain TE, they would optimize without payingattention to how the coupling flows affect the other ISPs op-timization problem. For example, is an output of ’soptimization problem and is thus under its control. However,

is an input to ’s optimization problem and thus affectsits outcome. Now, if is myopic, it will optimize withoutpaying attention to how much it may be hurting by de-termining myopically. Similarly, could determinemyopically. Thus, in this process, both ISPs may end up hurtingeach other.

When both ISPs route myopically, we denote the ISPs utilitiesas and , respectively.5 The question then arises:Is there any way that the ISPs could somehow cooperate ondetermining the coupling flows and improve their performance,i.e., achieve and such that:

1) the gains from cooperation are equitable (or fair) whileoperating at a Pareto-efficient operation point?

2) ISPs don’t have to divulge any critical information abouttheir networks?

The answer to the first question lies in the idea of Nash Bar-gaining [10]. The answer to the second question lies in the ideaof decomposition [12]. We explain both of these ideas next.

B. Nash Bargaining

The basic idea behind Nash bargaining is as follows. We as-sume that the ISP utilities are interdependent, concave (in net-work flows), and cardinal, where by cardinal we mean that theactual values of utilities matter—as opposed to ordinal utilitieswhere only the relative ordering of outcomes matters. Average

4We make the notation precise in Section III-C1.5In this paper, hot-potato routing and Nash equilibrium routing are taken as

representative myopic routing strategies. More details on these two strategiesfollow later in this section.

Fig. 2. The feasible region with Pareto-efficient frontier.

network delay as well as maximum link load are both examplesof (convex) cardinal utilities. Fig. 2 shows a typical feasible re-gion for two concave and cardinal utilities, where the feasibleregion is defined as the region where both ISPs would do betteroff compared to the myopic outcome. The myopic outcome isalso referred to as the breakdown point.

A fair and Pareto-efficient outcome, also referred to asthe Nash solution, can be obtained by maximizing the Nashproduct given by . Using the axiomatic theory of coopera-tive games, it can be shown that when two players (ISPs for us)with equal market power bargain, using threat strategies, theyshould arrive at the Nash solution.6 Referring to Fig. 2, thesethreat strategies correspond to the breakdown point, which isthe outcome if the ISPs are unable to reach an agreement.

In what follows, we provide a brief summary of the propertiesof the Nash solution, using the axiomatic bargaining approach.The idea here is that a good bargaining solution should satisfythe following four axioms, which we simply state as follows (see[10] for a detailed discussion).

1) Pareto Efficiency: This is obviously desirable since ISPsprefer more to less.

2) Symmetry: This says that the solution should provideequal gains from cooperation when the feasible region is sym-metric, where by symmetric we mean that the feasible regionis agnostic of the player’s identities and that it would look thesame even if the ISPs utility axis were swapped.

3) Independence of Affine Transformations: This requiresthat the solution should be agnostic of any affine transforma-tions (that is, shifts and scalings) applied to any of the two utili-ties. Therefore, if the solution is given by for someutilities , and is scaled and shifted to , thenthe solution should change to .

4) Independence of Irrelevant Alternatives: This basi-cally says that addition of irrelevant alternatives should notchange the solution. That is, for feasible regions and , if

, , andthen .

It turns out that the Nash solution is the only solution thatsatisfies these four axioms [9]. In fact, the Nash solution is the

6The axiomatic theory does not worry about how players reach a solution.A different form of game theory, referred to as noncooperative game theory,focuses on such dynamic issues. We leave the examination of such dynamics forfuture work. Also, in what follows, we have assumed that the ISPs have equalbargaining power. The Nash solution can be easily modified to reflect differentbargaining powers [10].


only solution that satisfies the following problem that is simulta-neously utilitarian (Pareto-efficient) and egalitarian (fair) [10].That is, the Nash solution effectively solves (overthe bounded set ) a modified problem

for some and that are endogenously determinedthrough solving .7 In particular, the Nash solution isproportionally fair [11]. This means that moving away from theNash solution causes a negative cumulative percentage changein utilities. That is, if is the Nash solution, and wemove to another point , then

(1)

We next describe how decomposition can be used to jointlyoptimize the Nash product without revealing any sensitive in-formation.

C. Decomposition

The idea of decomposition is not new. It has been success-fully used to solve large scale optimization problems [12] andto solve separable problems in a decentralized manner. More-over, in our case, decomposition allows separate entities in theoptimization problem to hide their internal critical information.In what follows, we first develop a precise optimization frame-work, and then use this framework to explain decomposition.

1) The Optimization Problem: We denote the networktopology of , , by a directed graphwith nodes and internal links. We alsodenote by the set of directed peering links. We then definethe incidence matrix for as matrix , with

if link leaves node , if link entersnode , and 0 otherwise.

We consider aggregate data flows through the network, wherewe identify each flow by its destination node. We denote bythe set of all possible destination nodes. For , we denote thenonnegative amount of flow originating at node and destinedto node by ( ). When , is the nega-tive sum of the flows destined to the node , thus ensuring flowconservation. We refer to as the source-sink vector.Note that include and , as described inSection III-A. Similarly, for , we denote the amount of non-negative flow destined to node on each internal link by

. We call the internal flow vector for destination .Finally, we denote the amount of nonnegative flow destined tonode on each peering link by . We call the

7We emphasize that the �’s are for exploratory purpose only and are auto-matically generated while solving �� . The modified problem is meantto illustrate the in-built egalitarian nature of the Nash bargaining solution. Thisscaling by �’s does not change the Pareto-efficient frontier in Fig. 2, i.e., thevalues of the choice variables resulting in Pareto-efficient points remain thesame. These�’s bring the usually incomparable� and� on a common footingso that we can talk about fairness in the first place.

peering flow vector for destination . This includes and, as described in Section III-A.

Now, we are ready to define the optimization problems invarious scenarios. We first present the Nash product problem,where the ISPs would jointly solve

(2)

for all , where the optimization variables are , , and. Here the two equality constraints are the flow conservation

constraints for and , respectively, and the last set ofinequality constraints ensures that the choice variables are non-negative. Note that other constraints such as link capacity con-straints can be readily included.

A related problem to (2) is when both ISPs route myopically.The myopic routing schemes that we are particularly interestedin are as follows.

Hot-Potato Routing: Under this approach, each ISP routesinterdomain traffic originating in its network to the closest (orminimum cost) peering point. For example, based on static costsassigned to links, the least OSPF-cost routing minimizes the costto route traffic on a per-source basis. If all links are assigned thesame weight, this routing results in routing traffic to the closestpeering points on a per-source basis. In a way, this attemptsto minimize the network resources consumed by interdomaintraffic within the source ISP network. This form of interdomaintraffic exchange is commonly used today.

Nash Equilibrium: Under this approach, ISPs myopi-cally optimize local objectives while iteratively playing bestresponses to each other. Each ISP finds the optimal way to splitinterdomain traffic across peering links, given the traffic splitsof its neighbor, until no better traffic split can be found. Thisdynamics eventually finds an equilibrium, also known as theNash equilibrium [8], from which no ISP has an incentive todeviate.

The major difference between the two schemes is that theNash equilibrium routing takes dynamic load-based costs onthe links into account as opposed the static weight-based op-timization of the hot-potato routing. Since the Nash equilibriumrouting is a dynamic process, it may take many iterations to con-verge. Hot-potato routing, on the other hand, finishes in one it-eration.

Under these routing schemes, each ISP myopically solves anoptimization problem

(3)

for and for all . Here, we use insteadof to represent the fact that the myopic routing strategieschange the original flow vectors as used in (2). For example, inhot-potato routing, since each ISP routes the interdomain flowsto the nearest exit points, the flow vector reflects some sources


of flows being on peering points instead of being on internalnodes belonging to the other ISP. That is, for optimizationproblem, represents not only the traffic that it wants to send(to its internal nodes as well as to ), but also the traffic sentby that would appear on its peering nodes due to the natureof hot-potato routing traffic that need to route eventually.In Nash equilibrium, where the ISPs iterate over the myopicproblems based on current incoming flows, the flow vector mayreflect a similar transformation.

2) Applying Decomposition to Nash Bargaining: Now welook into how we can cast (2) in separable form, allowing fora decentralized solution. We face two challenges: first, as itstands, the objective is not separable, and second, the ISPs util-ities are coupled through .

We first transform (2) into an equivalent problem by takingthe logarithm of the objective function. Since the logarithm isan increasing and concave function, this transformation does notchange the solution to the original problem [13]. We then get thefollowing equivalent problem8

(4)

We next introduce new nonnegative variables and ,which are local versions of for and , respectively.Problem (4) can then be rewritten as

(5)

We still have a coupling constraint , which we deal withusing dual (or pricing) decomposition [12], which is outlinednext.

We first write the partial Lagrangian of (5), with respect tothe coupling constraint, as

where are Lagrange multipliers associated with thecoupling constraint. This is a separable function in and

. We now solve the dual problem of (5), given by

(6)

8Remark: We note that taking logarithms requires � � � and � � �. Onehas to be careful while transforming an original cost minimization problem intoa utility maximization problem. That is, though �� isequivalent to �� , taking logarithm of �� isnot acceptable. Thus, we have to ensure that �� over the feasible region.We do so by setting �� , where � � � ��: i.e., � is a largepositive number.

where is given by the optimization problem

(7)

and is given by the optimization problem

(8)

We note that the optimal value of the dual problem (6) willbe equal to the optimal value of the primal problem (5) since(5) is a convex problem with a strictly feasible point, and strongduality holds by Slater’s condition [13]. We can readily recoverthe optimal flow allocation from the solution of the dual problemby ensuring, using a small regularization term, that the objectivefunctions are strictly convex (or concave) [13, §5.5.5].

The dual problem (6) is also referred to as the masterproblem. We can solve the master problem using various itera-tive methods. We choose the subgradient method [30] since itrequires very little coordination between (7) and (8) and allowsencapsulation of the internal data.

The subgradient method requires subgradients of and .A subgradient of is evaluated as follows. We first find and

that minimize

over and . Then, a subgradient of at is given by. Similarly, a subgradient of at is given by . Thus,

a subgradient of the dual function is given by, which is nothing more than the consistency check for

the coupling constraint.Dual decomposition, using the subgradient method for

solving master problem, then gives the following algorithm.repeat

1. Solve the subproblems (7) and (8). Obtain , .2. Update master (6) subgradient: .3. Update master (6) prices: .

Here, is the step size at the th iteration. We use a constantstep size, which guarantees convergence to an -ball around theoptimal solution, e.g., see [30] for more details. The subgradientmethod does not have a good stopping criterion, and in practiceit is often terminated when there is no additional progress in theminimization.

We note the following about the proposed procedure. First,the subproblems in Step 1 are independent and can be solvedby the ISPs independently of each other. Thus, we achieve ourobjective of not revealing critical information about the internalnetworks. Second, the updating of the Lagrange multipliers inStep 3 can happen in many ways. One way is for the ISPs to


announce the local versions of the coupling flows, i.e., and. Now, they can both calculate the new Lagrange multipliers.

Third, in our simulation experiments over realistic ISP topolo-gies, we find that this process typically converges in 50–100 it-erations to well within the optimal solution using a fixed stepsize.

IV. PRACTICAL ISSUES

In this section, we look at implementation issues related to theapproach developed. We emphasize that this section is intendedto work as a guide and is not a comprehensive design effort.

A. Implementation and Deployment

We observe that the easiest path to the adoption of our ap-proach is when individual ISPs employ it in conjunction withcentralized routing platforms such as rcp [31] or 4d [32]. In thissetup, the centralized routing controller of an ISP executes theprotocol in conjunction with the controllers of its neighbors. Thecontrollers exchange prices and negotiate flow splits in an of-fline manner until they reach the negotiated solution; this canbe done either through the network control plane or even via aseparate network socket. For example, they can set up a TCPconnection and constantly change information over this con-nection. Each controller then converts the negotiated solutioninto appropriate forwarding table updates on ISP routers. Notethat the forwarding tables are not updated with intermediate so-lutions. Furthermore, given the communication and computa-tional complexities in Sections IV-B and C, our solution wouldtake at most a few minutes to compute the solutions in responseto traffic matrix changes, which happen relatively infrequently.While it may be possible to implement our approach in a com-pletely distributed manner (e.g., where individual routers par-ticipate in negotiation), we believe that the above approach is asimpler alternative.

Given the complexity of current computer networks, it is pos-sible that rcp and 4d themselves may not get deployed, and evenif they do, it may take some time. However, it does not mean thatour approach cannot get deployed until ASs implement thesecentralized protocols. It is possible to use our approach in con-junction with traditional intradomain approaches such as OSPF.The final (not the intermediate) flow splits coming out of our op-timization programs can be converted to OSPF weights with theobjective that the resulting flow splits from using OSPF resultsin a utility close to the optimum. This is known to be an NP-hardproblem, requiring the application of heuristic approaches [21].The local search approach outlined in [21] works well in manypractical scenarios, however, there are many realistic situationswhere the performance gap from the optimum is significant [21].This may happen due to two reasons: first, OSPF protocol doesnot split traffic evenly on multiple shortest paths, and second,in many cases it is computationally intractable to find the bestlinks weights under OSPF. A recent work [33] shows that theperformance gap can be closed further by allowing routers todirect traffic on nonshortest paths, with an exponential penaltyon longer paths.9

9We note that the inputs to our approach come from traffic matrix estimationalgorithms and the outputs of our approach go to shortest path computation ones.Both of these may involve significant errors. However, our approach does notadd any errors of its own—it works in a transparent manner.

B. Communication Complexity

The master problem (6) in our approach is solved using thesubgradient method which typically takes 50–100 iterations toconverge (see [34, section 10.3.2]). Denoting the number of sub-gradient method iterations as , the number of peering links as

, and the maximum number of flows as , whereare the number of nodes in and , re-

spectively, we need to communicate real numbersper iteration, or real numbers total. Thesecan be converted to bits assuming (typically 32 or 64) bitsper real number. That is, our approach would require a total ofabout 8 MB of communication for a pair of ISPs with a totalof 500 nodes and 20 peering links—a very reasonable numbergiven the high capacity of today’s networks.

C. Computational Complexity

The subproblems (7) and (8) in our approach are solved usinginterior-point methods [13], [34]. Theoretically, these methodshave polynomial complexity in the number of variables , i.e., in

, where and are as defined in theprevious subsection and are the number of internallinks in and , respectively. In practice, however, sincewe can exploit inherent structure in our problems, these methodscan be efficiently implemented to solve the subproblems evenfor large ISP networks with thousands of links and destinations.For example, our Matlab implementation takes about 1 s to solvea subproblem with links and destinations,which translates to 10 000 variables.

V. EXTENSIONS

In this section, we discuss simple extensions to our basicframework that show: 1) how our framework can apply to mul-tiple peering ISPs; 2) how to accommodate single link failures;and 3) how to react quickly under arbitrary failures and changesin traffic demands. We emphasize that, similar to Section IV,this section is intended to work primarily as a guide and is notcomprehensive by any means.

A. Multi-ISP Extension

So far, our discussion has focused on a pair of neighboringISPs. How do we extend this to a more realistic setting with mul-tiple ISPs (note that peering is always in a pairwise manner)? Wenote that this is a nonissue if peers are not used for transit—thatis, for carrying traffic that is not destined to nodes in their ownnetworks. In that case, our basic framework simply applies pair-wise to multiple ISPs.

It becomes more interesting when peers can be used fortransit. This can be arranged through explicit transit agree-ments. Though we do not provide a complete theory, we doprovide an illustrative example. Consider three ISPs— ,

, and —with the agreement that can send trafficto destinations in either via direct peering links or through

(as transit) (Fig. 3).A scheme that provides flexibility in routing its traffic

to is as follows. Under this scheme and agreea priori on the total traffic, denoted by , to thatwould be allowed to route through . Note that this is con-sistent with how transit contracts are drawn today: A transit ISP


Fig. 3. The three-ISP model.

basically agrees to carry some traffic and charges for it on aper-Mb/s basis. Thus, this scheme conveniently allows the use ofbilateral interconnection agreements [35] that ensure end-to-endconnectivity in the Internet.

Under this scheme, using a direct extension of the dualdecomposition approach described in Section III-A, the ISPsjointly optimize the Nash product [or equivalently,

] with the constraint, which we refer to as thetransit flow constraint, that the total traffic destined to nodesin on the peering links between and (as wellas on the peering links between and ) equals .This constraint is in addition to the usual flow conservationconstraints for each ISP: first, all the traffic sourced from nodeswithin the network must be routed, and second, all the trafficdestined to nodes within the network must be routed. Thus, wesee that our original approach extends to the three ISP case in afairly straightforward way.

In fact, this approach can be extended to any arbitrarytopology in a similar way, as follows. In case of multiple ISPson a path between a pair of source and destination ISPs, therewill be multiple transit agreements, with one transit agreementper pair of neighboring ISPs.10 Then, each transit ISP will havea transit flow constraint corresponding to each transit pipe (i.e.,total traffic for the transit connection).

B. Making the Approach “Incremental”

In the general case, ISPs would run a protocol based on ourapproach at certain times of the day in a somewhat semistaticmanner (this might correspond with the time granularity atwhich demand information is collected). However, traffic de-mands may change suddenly due to phenomena like denial ofservice attacks, flash crowds, etc. In addition, links may fail,requiring the ISP to reroute its traffic. How can we extend ourapproach to react to incremental changes in internal topologiesand in traffic demand? We look at various scenarios.

1) Single Link Failures: We now show how our frameworkcan deal with single link failures in real time, eliminating theneed for dynamic renegotiation of flow splits. ISPs can iden-tify small lists of links that may fail with high probability (e.g.,planned outages or based on historical data). Say andidentify and number of links, respectively. Assuming

10This is how interconnection agreements are drawn today—always on a bi-lateral basis [35].

the probability of simultaneous multiple link failures to be verysmall, only a single link would fail at a time in either of thedomains. Thus, the ISPs would need to run the basic algorithm

times—once for the default (no failure) case andtimes to cover each of the single link failures—and

store the resulting flow splits for each. These could be indexedusing previously agreed upon unique keys. Upon failure of oneof these links, the ISP with the failed link can notify the otherISP that a link has failed and that they need to switch to the flowsplits corresponding to the failure. They can then switch to thenew flow splits as soon as possible, preferably in a coordinatedfashion. There is no need for renegotiation.

2) Dealing With Arbitrary Failures or Changes in Demand:We now show how to accommodate any link failure and sig-nificant shifts in traffic demands. Say a link fails in . Howshould the ISP modify its local routing to deal with this, whileminimizing its own cost and not impacting the other ISP ad-versely? A similar issue arises when traffic volumes for certainingress–egress pairs change suddenly (e.g., due to flash crowds).

The answer lies in using the equilibrium shadow pricesarising out of our approach. These indicate how expensive (orcheap) it would be for the receiving ISP if the correspondingflow splits are changed. When sudden internal changes occur,the sender can use this information to check if local changesmay be performed in order to accommodate the change and yetnot impact the neighbor in any significant manner. If the impactis likely to be low, the sender can make local changes. If theimpact is likely to be high, the sender will have to renegotiateor bargain the prices.

C. A Note on Incentive Compatibility

It is well known that Nash Bargaining is not incentive com-patible [36]. Therefore, our approach is susceptible to cheating.However, we believe that this will not hinder ISPs from adoptingour approach, for three reasons. First, we believe that the realworld is more cooperative than often depicted in the noncoop-erative game theory setting, and ISPs are honestly trying to im-prove their performance. Second, since our approach guaran-tees that ISPs see nonnegative improvement when compared totheir default strategies (e.g., hot-potato routing or Nash equilib-rium), a cheating ISP may be able to gain unfair advantage butit cannot degrade the performance of an honest neighbor com-pared to the neighbor’s default (that is, the breakdown point).Finally, it is possible to use other means to deal with potentiallycheating ISPs. For example, an interconnection contract may belegally binding and a cheating ISP may face the risk of losingit if caught. Nevertheless, we recognize that addressing incen-tive compatibility would make our work more complete, and wehope to address it in future work.

VI. EXPERIMENTAL RESULTS

We conduct simulation experiments to evaluate our approach.As mentioned earlier, our focus is on a pair of neighboring ISPs.Though it would be nice to include multi-ISP simulations, wefeel that the two-ISP simulations validate our approach, and in-cluding a multi-ISP extension would not add much value. Fur-ther, we are limited by space.


Fig. 4. Nash bargaining compared with hot-potato routing. (a) Load/Load. (b) Latency/Load. The top (bottom) graph plots the percentage improvement in theoptimization objective for �� (�� ) on the y-axis for each ISP pair. The ISP pairs are shown on the x-axis.

We use Rocketfuel ISP maps which contain PoP-level con-nectivity information [37]. The links in each map are annotatedwith the propagation latency, as well as the inferred OSPF linkweights employed by the ISP for its internal routing [38]. Themaps also include information on the peering locations of neigh-boring ISPs.

Two key components which are missing from the maps arethe traffic demand matrix (both intra- and interdomain), and thelink capacities. For the former, we use gravity-based models[39] where the demand between a pair of cities (or PoPs) isproportional to the product of their populations (the populationscan be obtained from public databases). Also, we assume thatthe demand matrix is symmetric. This model applies to bothintra- and interdomain traffic.11

For the latter, we assume that all links in an ISP have thesame capacity, where the capacity is computed as follows. Weuse the ISP link weights to compute the best routes betweendifferent PoP pairs (breaking ties randomly). We then identifythe most heavily loaded link and set its capacity to be twice thetotal demand carried by it. We use the same capacity for all otherlinks. We tried with other link capacity assignments (such asa bimodal distribution), but found the results are qualitativelyunchanged.

We simulate two different ISP optimization objectives:1) Load: the goal of the ISP is to minimize the maximum

link load in its network. Here, the load of a link is thetraffic volume it carries divided by its capacity.

2) Latency: the goal of the ISP is to minimize the maximumlatency incurred by the traffic it carries. When a trafficdemand is split between multiple paths, we compute theweighted latency for the demand-split, where the weightis simply the fraction of the demand routed on a path.

Note that both of these objectives are concerned only with in-tranetwork characteristics. For example, while optimizing forlatency, the ISPs only optimize for average delay within theirrespective networks, as opposed to end-to-end delay [21].

Our simulations compare the Nash bargaining approach withtwo other approaches: hot-potato and Nash equilibrium are my-

11We use a lower proportionality constant for interdomain traffic.

opic routing approaches described in Section III-C1. In our sim-ulations, the Nash bargaining outcome is calculated using themyopic policy as the breakdown point.

A. Nash Bargaining Versus Hot-Potato Routing

In Fig. 4, we compare the performance of our approachagainst the case where the ISPs employ hot-potato routing.Here, we consider two situations: one where both ISPs employthe same utility—Load—shown in Fig. 4(a), and the otherwhere employs Latency while employs Load,shown in Fig. 4(b). We make the following observations.

First, the objective of either ISP always improves, irrespectiveof whether the ISPs are optimizing similar objectives or not.In some cases, the value of the objective for one of the ISPsimproves twofold; this can be observed in both Fig. 4(a) and (b).In other cases, both ISPs see greater than 50% improvementeach. These results show that, in practice, peering ISPs can bothgain significantly from shedding their unilateral TE approachesand adopting the cooperative Nash bargaining-based approachwe propose.

Second, we note that the percentage improvements for thetwo ISPs are not necessarily equal. In some cases, one ISP gainssignificantly while the other sees no improvement at all aroundISP-pair 150 [see Fig. 4(a)]. This effect is especially pronouncedfor Fig. 4(b), where the ISP utilities are different. The asym-metry in the gains arises due to two reasons:

1) Given that the latency objective is an average (it cancelsout deviations around the mean) as opposed to the loadobjective, which is a maximum (it accentuates differ-ences), the default strategy (hot potato) may already beoffering fairly good performance for the latency objec-tive. Nash bargaining offers incremental benefits. Thisis in agreement with the observations in [6].

2) When utilities are dissimilar and, therefore, not directlycomparable, we cannot expect identical percentageimprovements anyway. Nevertheless, as mentionedin Section III, our approach offers proportional fair-ness, a highly desirable property. We illustrate this inSection VI-D.


Fig. 5. Nash bargaining compared with hot-potato routing. (a) Load/Load. (b) Latency/Load. The top (bottom) graph plots the fraction of ISPs with below %improvement for �� (�� ) in the ISP pair. The % improvements are shown on the x-axis.

Fig. 6. Nash bargaining compared with Nash equilibrium. (a) Load/Load. (b) Latency/Load. The top (bottom) graph plots the percentage improvement in theoptimization objective for �� (�� ) on the y-axis for each ISP pair. The ISP pairs are shown on the x-axis.

Fig. 5 shows the simulation results in the form of cumulativedensity functions (cdfs). These again demonstrate the signifi-cant improvements in case of the load objective, and the minorimprovements in case of the latency objective.

B. Nash Bargaining Versus Nash Equilibrium

In Fig. 6, we compare the performance of our approachagainst the case where the ISPs are in Nash equilibrium. Asexplained above, Nash equilibrium arises when each ISPsoptimizes its local objective while playing best responses toits neighbor.12 We note that the Nash equilibrium reflects thebehavior of selfish (myopic) and smart ISPs, while hot-potatois a greedy strategy. However, unlike hot-potato routing, theNash equilibrium may be hard to realize in practice, sinceconvergence in finite time is not always guaranteed. In oursimulation of Nash equilibrium, we simply ignore cases wherethe equilibrium is not found after a threshold amount of time.

12A brief note on the existence and uniqueness of the Nash equilibrium isin order. Given that we are looking at continuous and convex utility functions(“maximum load on a link” and “average delay”), from [10], a Nash equilibriumalways exists. The uniqueness property is a much harder objective, and we donot focus on it in this paper.

Here, we again consider two situations: one where both ISPsemploy the same utility—Load—shown in Fig. 6(a), and theother where employs Latency while employs Load,shown in Fig(b). As with hot-potato routing, we note that Nashbargaining offers superior performance to both ISPs when com-pared to the performance at the Nash equilibrium. This improve-ment is more obvious in Fig(b). This further establishes the ben-efit of bilateral cooperation in interdomain traffic engineering.

C. Nash Equilibrium Versus Hot-Potato Routing

We note an interesting fact from Figs. 4 and 6: Though hot-potato routing performs better than Nash equilibrium for a neg-ligible fraction of cases, in the majority of cases, the perfor-mance of both the ISPs at Nash equilibrium is better than thatwith hot-potato routing. This points to the fact that even amongmyopic unilateral approaches, the commonly used hot-potatorouting is not the optimal!

D. Efficiency and Fairness

We illustrate the fact that our approach yields a Pareto-effi-cient and fair solution using a spot-study of a randomly chosenpair of peering ISPs with AS numbers 1 and 5650. has


Fig. 7. Feasible region with Pareto-efficient frontier. The utilities shown cor-respond to � � �, where � is some large constant, and � is the value of theLoad objective. This transformation does not change the guarantees offered byour solution.

110 bidirectional links and 42 nodes, whereas has 54 bidi-rectional links and 22 nodes. In addition, there are four bidirec-tional peering links. Fig. 7 shows the feasible region (shadedgray) when both ISPs employ the Load utility. It also shows theindifference curves for as well as the Nash equilibrium andhot-potato points.

Our Nash bargaining approach finds the unique solution de-noted by point , with . It canbe easily verified that this point is Pareto-efficient as well as pro-portionally fair.

VII. SUMMARY

In this paper, we presented a new interdomain traffic en-gineering approach that is Pareto-efficient, fair and does notrequire ISPs to reveal internal information. Our approachuses ideas from cooperative game theory (specifically, Nashbargaining) as well as a host of techniques from nonlinearoptimization (such as dual decomposition and the subgradientmethod) to achieve the above desirable properties.

We simulated our approach over real ISP topologies andtraffic demands. We found that our approach can offer signif-icant improvement both relative to prevalent approaches suchas hot-potato routing, as well as more sophisticated selfishinterdomain TE approaches. We also empirically verified thefairness and efficiency properties of our approach.

Our solution provides provable guarantees that are missingfrom the state-of-the-art in interdomain traffic engineering.Therefore, our approach is very amenable to adoption by ISPstoday.

ACKNOWLEDGMENT

The authors would like to thank Prof. R. Johari andProf. S. Boyd for their guidance.

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Gireesh Shrimali received the B.Tech. degree fromthe Indian Institute of Technology, Delhi, India, in1991; the M.S. degree from the University of Min-nesota, Minneapolis, in 1993; and the Ph.D. degreefrom Stanford University, Stanford, CA, in 2008.

He is an Assistant Professor with the Indian Schoolof Business, Hyderabad, India. Between receiving theM.S. and Ph.D. degrees, he worked in the industry forclose to nine years. His current research interests arein design and analysis of high-speed, high-capacitynetworks. His research also focuses on networks and

Internet economics.

Aditya Akella (M’06) received the B.Tech degreein computer science and engineering from the IndianInstitute of Technology, Madras, India, in May2000, and the Ph.D. degree in computer sciencefrom Carnegie Mellon University, Pittsburgh, PA, inSeptember 2005.

After spending a year as a Post-Doctoral Fellowwith Stanford University, Stanford, CA, he joinedthe Department of Computer Sciences, Universityof Wisconsin, Madison, as an Assistant Professorin Fall 2006. His research interests include internet

routing, network protocol design, Internet security, and wireless networking.

Almir Mutapcic (M’97) received the B.S. degreein electrical engineering from the University ofMissouri, Rolla, and the M.S. and Ph.D. degreesin electrical engineering from Stanford University,Stanford, CA, in 2002 and 2008, respectively.

His research interests include applications ofdistributed optimization and robust convex optimiza-tion.

cooperative interdomain traffic engineering using nash bargaining and decomposition

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