THE primary responsibility of an Internet Service Provider. Optimal Configuration for BGP Route Selection

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5 since traffic for P 2 from A 1 and A 2 egress at routers b 3 and b 4, respectively, which are closest to the respective ingress routers for the traffic. Note that for the multiple egress case, if hot potato routing is used, A 2 will try to send traffic for both prefixes to router b 4. However, b 4 only has capacity 50, and so will not be able to accommodate all the traffic from A 2. B. Integer Program Formulation The BGP egress selection problem can be formulated as an integer program. For each prefix P k and the traffic from a neighbor A h ingressing at edge link b i intended for that prefix, we must select an egress edge link b j such that j Out(k). Define variable x j hik to denote this selection, so xj hik =1if b j is selected as the egress edge link for traffic from b i to prefix P k, and x j hik =0otherwise. The integer program, IP(1), for the BGP egress selection problem is then formulated as follows: min x j hik d(i, j) t(h, i, k) (1) k h i In(h) j Out(k) subject to the following constraints j : x j hik t(h, i, k) C j (2) k:j Out(k) h i In(h) h, i, k : j Out(k) x j hik =1 (3) The objective function of (1) is the integer programming version of the minimization objective noted for both problem variants in the problem statement. Constraint (2) enforces the egress capacity constraints in the integer program. Equation (3) is used to specify that any traffic from a particular neighbor, ingressing at a particular edge link, and destined for a particular prefix, must go through one selected egress edge link toward the destination prefix. The formulation as presented thus far allows for multiple egress edge links for a given prefix across different neighbors, but does not enforce the proximity constraint. In order to further constrain the formulation, we define an additional set of integer variables z j k such that zj k = 1 if b j is chosen as an egress point for traffic to prefix P k for one or more ingress neighbors, and z j k =0otherwise. Then, the following equations enforce the proximity constraint. h, k, i, j : x j hik zj k (4) h, k, i, j : x j hik,zj k {0, 1} (5) h, k, i In(h), j Out(k),j Q(i, j, k) : z j k + xj hik 1 (6) In Equation (6) above, Q is a utility function that is used to specify, for a given ingress and egress edge link pair b i and b j and a given prefix P k, the set of alternative egress edge links for P k that are closer than b j. Thus Q(i, j, k) is defined as the set of edge links {l l Out(k) d(i, l) <d(i, j)} 4. This final 4 In case d(i, l) =d(i, j), then l Q(i, j, k) if and only if the IP address of l is smaller than j. constraint (6), which is the proximity constraint, ensures that an (A h,b i,p k ) triple cannot be assigned an egress edge link b j if there exists a h or i such that (A h,b i,p k ) is assigned to an egress edge link b j, j Q(i, j, k). This completes the integer programming formulation for the multiple egress selection. To change the formulation for the single egress selection case, we simply further constrain z j k. To assure that for a prefix P k, only a single egress is selected across all edge links, we replace Equation (6) by the following: k : j Out(k) z j k =1 (7) Unfortunately, solving the above integer program IP(1) is known to be computationally intractable. However, in Section IV, we show that solving the linear relaxation of a variant of the above integer program enables us to derive efficient solutions for the MES problem. III. Algorithms for Single Egress Variant The SES problem can be shown to be a version of the generalized assignment problem (GAP) which is known to be NP-hard and has been well-studied in the operations research and theoretical computer science communities. See for example, [13] and [14]. The definition of GAP is as follows. Generalized Assignment Problem: Given ξ jobs and φ machines, a processing time p rs and cost c rs for processing job r on machine s, and a total processing time T s available for each machine s, compute an assignment f : {1,...,ξ} {1,...,φ} of jobs to machines such that the total cost of processing jobs is minimized, that is, r c rf(r) is minimum, and the processing time for jobs on each machine s does not exceed T s, that is, r:f(r)=s p rs T s. We map the SES problem to GAP by considering each egress edge link b j to be a machine and each prefix P k to be a job with processing time and cost h i In(h) t(h, i, k) and h i In(h) d(i, j) t(h, i, k), respectively, on machine j. The constraint on the processing time available on each machine b j is the capacity constraint, C j, of the egress edge link. Note that this is not the most general version of GAP as the processing times of the jobs do not vary with the machines (egress edge links). GAP is not only intractable, but also very difficult to solve approximately. Even ignoring costs (e.g., setting all job costs to 0), it is intractable to compute an assignment of jobs to machines such that the total processing time constraints of machines is not violated [15]. The version described above where the processing time of a job does not vary with the machines is also intractable [16]. Thus, the only option available is to rely on heuristics to solve GAP.

6 75 P1,P2 P1,P2 3 Egress 4 50 A1,A2 >P1 A1,A2 >P1 A1,A2 >P2 A1 >P2 A2 >P Egress Egress Ingress 2 A1,A2 A1,A2 (a) AS Ingress/Egress Routers 1 Ingress 2 A1 >P1,P2 A2 >P1,P2 (b) Optimal Assignment (Single Egress) 1 Ingress 2 A1 >P1,P2 A2 >P1,P2 (c) Optimal Assignment (Multiple Egress) Fig. 1. Example of optimal assignments for single and multiple egress cases A. Generalized Assignment Problem Heuristic As mentioned above, there is a lot of previous work on GAP and good polynomial-time approximation algorithms for GAP that relax machine processing time constraints exist in the literature. Let C be the cost of the optimal solution to GAP that does not violate any capacity constraints. Then an (α, β) approximation algorithm for GAP is one that gives a solution with cost at most αc and with capacity constraints violated by at most a factor of β. The best known result for GAP is by Shmoys and Tardos in [14] where a (1, 2) approximation algorithm is given. Lenstra, Shmoys, and Tardos [15] have also shown that it is NP-hard to obtain a (1,β) approximation algorithm for GAP for β<3/2. The algorithm of Shmoys and Tardos in [14] is based on first solving the LP relaxation of the integer programming formulation for GAP, and then rounding the fractional solution to a nearby integer solution. The total cost of the assignment computed by the Shmoys and Tardos algorithm is optimal, but, as noted above, the processing times of jobs assigned to a machine may exceed the machine s total processing capacity by at most a factor of 2. In Figure 2, we present a greedy heuristic that uses the assignment f computed by the Shmoys Tardos algorithm as the basis to compute a new assignment f that satisfies processing time constraints and still has a low cost. In Procedure Greedy, jobs on machines whose processing times constraints are violated are re-assigned to other machines with sufficient capacity to process the jobs. This process of rescheduling jobs from violated machines to non-violated machines is continued until either no violated machines remain or no more jobs on violated machines can be re-assigned. The critical issue is which jobs on violated machines should be chosen for migration and which machines should they be migrated to. We adopt a greedy approach, choosing during each iteration, the job r and machine t for which transferring r to t results in the smallest increase in cost per unit decrease in the violation amount. Note that in Step 11, c rt c rs is the increase in cost associated with re-assigning r from s to t, while min{p rs,u s T s } is the decrease in the violation. The time complexity of Procedure Greedy is dominated by procedure Greedy(c[], p[], T []) 1. Use Shmoys Tardos Algorithm to compute assignment f of jobs to machines for costs and processing times c[], p[] and T [] 2. foreach machine s, let U s := s:f(r)=s prs 3. Let VSet := {s : U s >T s} 4. job := 1 5. while job > 0 { 6. job := 0 7. min cost := 8. foreach job r such that f(r) =s and s VSet 9. foreach machine t not in VSet 10. if T t U t >p rt{ c rt c rs min{p rs,u s T s} 11. cost := 12. if cost < min cost{ 13. min cost := cost 14. job := r 15. old := s 16. new := t 17. } 18. } 19. if job > 0 { 20. f(job) :=new 21. U old := U old p job,old 22. U new := U new + p job,new 23. if U old T old then VSet := VSet - {old} 24. } 25. } 26. return f Fig. 2. Greedy Heuristic for GAP the running time of the Shmoys Tardos Algorithm in Step 1. The Shmoys Tardos algorithm rounds the fractional solution of the LP for GAP by finding a minimum-cost integer matching of a bipartite graph containing ξ job nodes on one side, at most ξ + φ machine nodes on the other, and no more than 2ξφ edges between the two sides. We solve the minimum cost integer matching problem using successive shortest path algorithm [17] for the minimum cost flow problem. This solution requires running the shortest path algorithm ξ times, with each run taking O((ξφ) log(ξ + φ)) time. Thus, the overall time complexity is O((ξ 2 φ) log(ξ + φ)). Note that ξ corresponds to the number of prefixes in

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