1. Multipath Routing Ph.D. Research Proposal Ron Banner Supervisor: Prof. Ariel Orda March 2004

2. Agenda u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Selfish multipath routing u Online multipath routing for congestion minimization u Future research

3. What is Multipath Routing? u Multipath Routing is the method of establishing multiple paths between given source-destination nodes within the network.

4. Advantages of Multipath Routing u Survivability  Provides redundancy. u Congestion avoidance  Improves network utilization.  Provides load balancing. u Management and control  Provides better performance in the presence of selfish/unregulated behavior

5. Previous Research u Survivability  Mainly solutions that focus on the establishment of pairs of disjoint paths (e.g., the 1+1 and 1:1 protection architectures). u Congestion avoidance  Mainly heuristics (e.g., ECMP).  Online: no previous work for multipath routing. u Management and control  No previous work on the degradation of network performance due to selfish behavior of users that employ multipath routing.

6. Notations G (V,E) – Directed Graph V - Collection of nodes E – Collection of links (edges) P (s,t) -Collection of all paths from s to t  (s,t) –flow demand from s to t d e -delay of link e c e -capacity of link e p e -failure probability of link e f e -flow rate on link e D(p) – the end-to-end delay of path p i.e., C(p) – the capacity of path p i.e.,  (p) – the reliability of path p i.e.,

7. Summary of results: Survivability u We provide a quantitative framework that specifies the desired level of survivability against single failures. c=20, p=0.05 c=30,p=0.05 c=10, p=0.05 c=30, p=0 c=30, p=0.05 S T

8. Summary of results: Survivability u We developed optimal polynomial schemes for 1:1 and 1+1 protection that consider important tradeoffs  Survivability vs. bandwidth  Survivability vs. feasibility.  … u No need to establish connections that consist of more than two paths. u Derived a new “hybrid” protection architecture that has several advantages over both the 1:1 and 1+1 protection architecture. u Show that by just slightly alleviating the requirement of full survivability a major improvement is obtained.

9. Summary of results: Congestion minimization-offline u Goal: Minimize network congestion when all demands are known in advance. u Cope with constraints  Delay jitter  End-to-end delay  Number of paths u Minimizing the congestion under end-to-end delay and/or delay jitter  NP-hard  Pseudo polynomial solution  optimal approximation scheme u Minimizing the congestion while restricting the number of routing paths  NP-hard  2-approximation scheme

10. Summary of results : Congestion minimization-online u Goal: Minimizing the network congestion when demands arrive one at a time. u Derived a multipath routing algorithm for congestion minimization with an O(logN)-competitive ratio. u Derived a lower bound of Ω(logN) for any online multipath routing algorithm for congestion minimization u Our algorithm is best possible.

11. Summary of results: Selfish multipath routing u Goal: Investigating the degradation in network performance due to selfish behavior of users. u Given a load-dependent performance function q e (f e ) for each link we consider bottleneck network objectives i.e., Max e  E {q e (f e )} and additive network objectives i.e., u Assume that users are selfish and their performance is dictated by their worst (bottleneck) elements. ∞ 1 ∞ M Additive Bottleneck Network objective Routing approach Multipath Routing Single-path Routing

12. Agenda u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Selfish multipath routing u Online multipath routing for congestion minimization u Future research

13. The tunable survivability concept u Current survivability schemes typically offer two degrees of protection against single failures  Full (100%) protection.  No protection at all. u In practice, the requirement of full protection is often too restrictive  In many cases it is infeasible (N. Taft-Plotkin, B. Bellur and R. Ogier).  In other cases it is very limiting (G. Maier, A. Pattavina, S. De Patre and M. Martinelli). u Tunable survivability enables to consider valuable tradeoffs.  Survivability vs. bandwidth  Survivability vs. feasibility  Survivability vs. end-to-end delay  …

14. Survivable connections u p-survivable connection: a collection of paths (p 1,p 2,…, p k )  P (s,t) ×P (s,t) ×…× P (s,t) that, upon a link failure, has a probability of at least p that at least one path out of (p 1,p 2,…, p k ) remains operational.  The bandwidth of a survivable connection with respect to the 1+1 protection architecture is the maximum B≥0 such that n·B ≤ c e for each link e that is common to n paths from (p 1,p 2,…, p k ). u The probability of a survivable connection to remain operational upon a single failure is the probability that all the common links are operational upon that failure i.e.,  The bandwidth of a survivable connection with respect to the 1:1 protection architecture is the maximum B≥0 such that B ≤ c e for each e that belongs to a path in (p 1,p 2,…, p k ).  It is also

15. Two Paths are Enough u Theorem Let (p 1,p 2,…, p k )  P (s,t) ×P (s,t) ×…×P (s,t) be a p-survivable connection. There exists a p-survivable connection that has at least the bandwidth of (p 1,p 2,…, p k ) with respect to the 1:1 (alternatively 1+1) protection architecture. u Proof (sketch for the 1:1 protection)  We shall construct only from the links that belong to paths in (p 1,p 2,…, p k ). Therefore, the bandwidth of is at least that of (p 1,p 2,…, p k ). u Formal proof Critical points

16. Most Survivable Connections with a Bandwidth of at Least B u Since two paths are enough, we focus on survivable connection that consist of two paths. u The most survivable connection with a bandwidth of at least B for the 1+1 protection architecture is established by a reduction to the min cost flow problem. u The flow demand is set to 2∙B flow units. A link in the original network Links in the transformed network Discard the link C e <B B≤C e <2∙B C e ≥2∙B c e =B, w e =0 c e =B, w e =-ln(1-p e ) c e,p e

17. Most Survivable Connections with a Bandwidth of at Least B u Since the flow demand and capacities are B-integral the min cost flow is B-integral. u The flow decomposition algorithm can be applied in order to decompose the B-integral link flow (that transfers 2·B flow units) into a flow over two paths: p 1, p 2 such that f(p 1 )=f(p 2 )=B. u Since the flow has a minimum cost, has a minimum value. u Therefore, (p 1,p 2 ) is a connection with a bandwidth of at least B that maximizes hence, it maximizes

18. Establishing Most and Widest p-survivable Connections u The most survivable connection is the connection that has the maximum probability to remain operational upon a failure  It is also the most survivable connection with a bandwidth of at least B=0. u The widest p-survivable connection is the p-survivable connection with the maximum bandwidth. u How to establish the widest p-survivable connection? u Idea : search for the largest B such that the most survivable connection with a bandwidth of at least B is a p-survivable connection. u It is enough to perform a binary search over the set  Why u The widest p-survivable connection is therefore established within O(logN) executions of any min cost flow algorithm.  Why

19. u The only difference in the reduction lies for the links that have capacities in the range [B,2B]. u For 1:1 protection only one of the paths carries B flow units. u Hence, all links that have a capacity in the range [B,2B] can concurrently be employed by both paths. A link in the original network Links in the transformed network Discard the link C e <B C e ≥B c e =B, w e =0 c e =B, w e =-ln(1-p e ) c e,p e Establishing Survivable Connections for 1:1 protection Go to 1+1 reduction

20. u The tunable survivability concept gives rise to a third protection architecture. u Reduces the congestion of all links that are shared by both paths w.r.t 1+1 protection. u Upon a link has a faster restoration w.r.t 1:1 protection. u Provides the fastest propagation of data. u However, requires additional nodal capabilities. The Hybrid protection architecture S T

21. u The hybrid architecture transfers through each link exactly one duplicate of the original traffic. u Hence, the bandwidth of (p 1,p 2 ) with respect to hybrid protection is u Hence, by definition, all schemes for 1:1 protection apply for hybrid protection. The Hybrid protection architecture Go to Def

22. Simulation results u We quantify how much we gain by employing tunable survivability instead of full survivability. u Random networks  10,000 Waxman topologies  10,000 Power-law topologies.  Explain the construction Bandwidth ratio (1:1)

23. Simulation results Bandwidth ratio (1+1) Feasibility ratio

24. u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Selfish multipath routing u Online multipath routing for congestion minimization u Future research Agenda

25. Problem formulation u Goals:  Minimize network congestion when all demands are known in advance  Cope with constraints (delay-jitter, delay, number of paths) u Performance Objective: network congestion factor  Minimizing  RFC 2702 and others.  No link becomes over-utilized.  More room for future traffic growth by maximizing the common scaling factor.

26. Requirements for practical deployment u Restricting the delay-jitter among all routing paths  RFC 2991.  Avoid the “fast retransmit” mode.  Reduce buffering requirements. u Limiting the number of paths per destination  S. Nelakuditi and Zhi-Li Zhang.  Reduce the tendency of packet reordering.  Reduce overhead.  Simplify the schemes that distribute traffic. u Bounding the end-to-end delay of each path.

27. Computational Intractability u Minimizing the network congestion factor under the end-to-end delay restriction is NP- hard.  Proof. u Minimizing the network congestion factor under the delay jitter restriction is NP- hard.  Proof. u Minimizing the network congestion factor under the restriction on the number of paths is NP-hard.  Proof.

28. Minimizing congestion while restricting the number of paths u Observation : The optimal network congestion factor of a  /K- integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K paths. u Proof: Let f * be a path flow that has the smallest network congestion factor α * among all path flows that transfers  flow units from S to T over at most K paths. f=2∙f* is a path flow with a network congestion factor 2∙α* that transfers 2  flow units from S to T over at most K paths. Round down the flow f(p) over each path to a multiple of  /K. Let f R be the resulting path flow. Given a network G(V,E) and a source- destination pair. Since f transfer 2  flow units over at most K paths f R transfers at least  flow units from S to T f R is a  /K - integral path flow that transfers at least  flow units from S to T and has a network congestion factor of at most 2∙ α *.

29. Minimizing the congestion under integrality restrictions u A  /K-integral path flow admits at most K paths. u Corollary: minimizing the congestion while restricting the flow to be integral in  /K is a 2-approximation scheme. u The network congestion factor of all  /K-integral path flows belong to.  The flow over each link is integral in  /K and is at most .  Hence, for each e  E it holds that  In particular,

30. Minimizing the congestion under integrality restrictions Goal: Find a  /K-integral path flow that has the minimum network congestion factor in Solution Find a path flow with the smallest such that the following procedure succeeds. u multiply all link capacities by a factor of α. u Round down the capacity of each link to a multiply of  /K.  Since the flow must be  /K-integral, such a rounding has no affect. u Apply a maximum flow algorithm that returns a  /K-integral link flow when all capacities are integral in  /K.  If the link flow transfers  flow units from S to T return Success  Else, return Fail

31. Minimizing the congestion under end-to-end delay restrictions - linear program u It is straight forward to extend the linear program to the multi-commodity case. u The path flow is constructed using a variant of the flow decomposition algorithm. u The complexity incurred by solving the linear program is polynomial in D  The number of variables is O(M·D).  The number of constraints is O(M·D).

32. Approximation Scheme u Goal: reduce the value of the end-to-end delay restriction D. u Delete from the network all the links with a delay d e >D. u Delay scaling: u Apply the linear program for the new instance.  As the new instance relax the original instance the congestion is not worse then the optimum. u Convert each non-simple path into a simple path. u Total error for a path: N· . u New end-to-end delay: D+ N·  =D∙(1+є).

33. Minimizing the congestion under delay-jitter restrictions u Idea: restrict the minimum end-to-end delay L and the maximum end-to-end delay U of the routing paths. u It is sufficient to add the linear program a minimum end-to-end delay restriction L.  New Linear Program. u Given a delay-jitter restriction J and an end-to-end delay D  For each L  [0,D-J] solve the new linear program with a minimum and a maximum end-to-end delay restrictions L, L+J, respectively. u Scaling down the end-to-end delay restriction D produces an є- optimal approximation scheme for the case where d max =O(J).  Details.

34. Agenda u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Selfish multipath routing u Online multipath routing for congestion minimization u Future research

35. Selfish Routing u Network users are selfish.  Do not care about social welfare.  Want to optimize their performance. u A central Question: how much does the network performance suffer from the lack of global regulation? u A flow is at Nash Equilibrium if no user can improve its performance.  May not exist.  May not be unique. u The price of anarchy: The worst case ratio between the performance of a Nash equilibrium and the optimal performance.

36. Previous Work u [Koutsoupias/Papadimitriou]  First paper to propose quantifying the cost of lack of regulation.  Concentrated on two node networks. u [Roughgarden]  General networks.  Infinite number of users.  users route traffic along the minimum latency path.  The price of anarchy is unbounded.

37. Model u A set of users U. u For each user, a positive flow demand  u and a source- destination pair (s u,t u ). u For each link e, a performance function q e (∙).  q e (∙) is continuous and increasing for all links. u Users behavior  Users are selfish.  They optimize bottleneck objectives u Network  Bottleneck objective  Additive objective

38. Non-uniqueness of Nash Equilibrium s t u One user wants to transfer 1 unit from s to t. u Assume that q e (f e )=f e for each e  E. u (f p1 =1, f p2 =0) & (f p1 =0, f p2 =1) are Nash flows with respect to unsplittable flow vectors. u (f p1 =0.5, f p2 =0.5) & (f p1 =0.25, f p2 =0.75) are Nash flows with respect to splittable flow vectors. u We identified two different Nash flow for each routing approach. e2e2 e1e1 e3e3 p1p1 p2p2

39. Existence of Nash Equilibrium  Definition: integral flow vector is a feasible flow vector where is integral in for each user u  U and p  P. u Theorem: Considering integral flow vector there exists a Nash equilibrium for each N  +.  The existence of NEP for Single-path Routing corresponds to the case where N=1.  The existence of NEP for Multipath Routing corresponds to the case where N →∞.  However, still needs to prove for the case where “N =∞”.  The proof of the theorem.

40. No price of anarchy for bottleneck network objectives u The price of anarchy is usually more than 1 and it is often unbounded.  Roughgarden: the price of anarchy is unbounded.  Papadimitriou: the price of anarchy is u Theorem: Given an instance [G(V,E), U,q e (·)]. If multipath routing is allowed then the price of anarchy is 1.  Proof. u Braess paradox: the addition of links to noncooperative networks can negatively impact performance of all users.  However, cannot occur for multipath routing (when q e (0)=0).

41. Price of anarchy is at most M with additive objectives u Theorem: Given an instance [G(V,E), U,q e (·)]. If multipath routing is allowed than the price of anarchy with respect to additive network objectives is M. u Proof:  Let f and f * denote a Nash and an optimal flow, correspondingly.  Therefore, B(f) ≤ B(f * ).  Therefore, max e  E {q e (f)} ≤ max e  E {q e (f * )}.  Hence, ∑ e  E q e (f)≤ M∙max E {q e (f)} ≤M∙max e  E {q e (f * )} ≤M∙∑ e  E q e (f * ) ■ u Corollary: Driving users to route traffic according to bottleneck metrics bounds the price of anarchy of additive network objectives to M.

42. Bad news for single-path-routing u The price of anarchy is unbounded for single path routing  Additive network objectives.  Bottleneck network objectives.  A =   B = 2∙  ST Additive Bottleneck Optimal flow Nash flow Price of anarchy

43. Agenda u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Selfish multipath routing u Online multipath routing for congestion minimization u Future research

44. The Model u Requests arrive one at a time and there is no a priori knowledge regarding future demands. u Each request specifies:  the source s r and destination t r.  the requested flow demand  r.  the maximum number of routing paths k r that can carry the demand. u Goal: Route all demands while minimizing the network congestion factor. u For the case were demands are limited to single an O(logN)- competitive strategy was derived by Aspnes, Azar, Fiat, Plotkin, Waarts.

45. Evaluating the Quality of Online Algorithms u A solution is offline if it is based on the entire input sequence. u The competitive ratio is the worst case ratio between the performance of the online algorithm and the performance of the optimal offline algorithm. u In our case the performance is the network congestion factor. u The entire requests sequence is denoted by R.

46. Minimizing the congestion under integrality restrictions u A path flow is  /K-integral if the flow of each request r  R over each path is integral in  r /K r. u Theorem: The optimal network congestion factor of a  /K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K r paths for each request r  R.  Proof.  A  /K-integral path flow employs at most K r paths for each r  R. u Corollary: minimizing the congestion while restricting the flow to be integral in  /K is a 2-approximation scheme.

47. Online solution u Upon the arrival of the n th request:  Split the request to K n successive requests to transfer  n /K n flow units.  Employ the online strategy of plotkin at el to route the demands over single paths. u Plotkin’s online strategy produces a competitive ratio of O(logN). u Therefore, we establish an online strategy with a competitive ratio of O(logN) for  /K-integral path flows. u Therefore, we establish an online strategy for our original problem with a competitive ratio of 2·O(logN)=O(logN). snsn  n /K n tntn

48. A Lower Bound of Ω(logN) for Multipath Routing S VNVN V N-1 V3V3 V2V2 V1V1 The K-th request wishes to transfer a flow demand of flow units from S to some target in layer K.

49. A Lower Bound of Ω(logN) for Multipath Routing (cont.) u After logN requests the network congestion factor is at least ½∙logN. u The optimal offline algorithm can achieve a network congestion factor of 1. S VNVN V N-1 V3V3 V2V2 V1V1

50. A Lower Bound of Ω(logN) for Multipath Routing (cont.)  There exists a lower bound of ½∙logN for networks with at most N’=N∙logN+N ≤2 N∙logN nodes. u We have to show that ½∙logN=Ω(logN’). u Indeed, there exists C>0 and N>N 0 such that logN’=logN+log(2·logN)=logN+log2+loglogN ≤ C∙ ½∙logN. There exists a lower bound of Ω(logN) for the best possible competitive ratio. Our online algorithm is best possible.

51. Agenda u Introduction & summary of results u Multipath routing schemes for survivable networks u Multipath routing schemes for congestion minimization u Online multipath routing for congestion minimization u Selfish multipath routing u Future research

52. Future research u Deepening the current work Deepening the current work u Selfishness in multipath routing Selfishness in multipath routing u Online multipath routing for finite holding time connections Online multipath routing for finite holding time connections u Other congestion criteria Other congestion criteria u Multipath routing and security Multipath routing and security u Recovery schemes for multipath routing Recovery schemes for multipath routing u Multipath routing and wireless networks Multipath routing and wireless networks u Fairness in multipath routing Fairness in multipath routing u Time dependent flow demands in multipath routing Time dependent flow demands in multipath routing

53. Deepening the Current Work u Consider for the proposed schemes  Distributed implementation  Heuristic schemes with low complexity  Multi-commodity extensions (congestion minimization)  Already considered in the scheme that restricts the end-to-end delay. u Establish a unifying scheme that bounds the number of paths, the end to end delay of each path, and the delay-jitter among all paths.  Online computation  Offline computation

54. Selfishness in Multipath Routing u In networks that have many users, the price of anarchy with respect to additive metrics may be very large. u If all users route their traffic with respect to bottleneck objectives, the price of anarchy with respect to additive network objectives is at most M. u Driving users to route traffic according to bottleneck metrics bounds the price of anarchy to M. u Advertising only the condition of the worst links may cause users to route traffic according to bottleneck metrics.  In that case, what can be said on the price of anarchy when the network manager advertises the condition of the K-worst links?

55. Online Multipath Routing for finite holding time connections u We have established an online strategy for permanent connections (i.e., connections with infinite holding times).  In practice, the holding times are usually finite. u There are online routing schemes with provable performance guarantees for the finite holding time case.  The holding time may be specified upon arrival  Only the distribution on the holding time is known.  No information on the holding time. u Investigate multipath routing for the finite holding time model.  Investigate the lower bound.  Establish corresponding multipath routing schemes.

56. Other Congestion Criteria u Thus far, we measured congestion according to the most utilized links in the network. u Although these links are the most severely affected by congestion, other links are affected as well. u Moreover, there are cases where congestion is better modeled through non-linear optimization functions. u Consider other optimization functions for congestion.  More general link congestion functions.  Already considered in the work on selfish routing.  Congestion functions that consider all the links in the network.

57. Multipath Routing and Security u Only the target sees the whole data stream when it is split among several node-disjoint paths. u Reconstructing the data stream is possible only at the target node. u It is essential to  Identify several node disjoint paths.  Assign a limited portion of the traffic over each path. u Develop multipath routing schemes that engage this inherent advantage  The solution must consider the requirements of multipath routing.

58. Recovery Schemes for Multipath Routing u Multipath Routing has the advantage of fast restoration upon a failure. u Upon a path failure, the data stream that traveled over the failed path may be split along the remaining paths.  Avoid additional path computation and resource reservation. u Requires that the sum of the spare capacities of the remaining paths is not smaller than the flow on the failed path. u Establish multipath routing schemes that enable fast recovery while considering the requirements of multipath routing.

59. Multipath Routing and Wireless networks u Energy Efficient Routing  In wireless networks nodes have a limited power resources (batteries).  Energy consumption is proportional to node transmission rates.  Therefore, splitting the traffic among several paths can prolong the time until the first battery is exhausted.  Establish schemes that maximizes the network’s lifetime while considering the requirements of multipath routing. u Survivability in wireless networks  Standard survivability schemes establish pairs of disjoint paths.  If two links that belong to different paths are too near, noise can affect both links.  Establish schemes that consider the minimum physical distance between two links that belong to different paths.

60. Fairness in Multipath Routing u A commodity may attempt to establish too many paths  In order to maximize its bandwidth.  In order to maximize its survivability. u This may come at the expense of other commodities.  E.g., a commodity may use too many entries in a (limited) routing table. u Seek suitable fairness criteria for multipath routing and establish schemes that incorporate the new criteria.

61. Time Dependent Flow Demands in Multipath Routing u We have assumed that flow demands are constant in time. u Often, flow demands are not constant.  Users send/receive data for short periods of time.  The TCP congestion control mechanism changes transmission rates with time.  Extend our model to cases where  →  (t).

62. The End

63. Two Paths are Enough u Theorem Let (p 1,p 2,…, p k )  P (s,t) ×P (s,t) ×…×P (s,t) be a p-survivable connection. There exists a p-survivable connection that has at least the bandwidth of (p 1,p 2,…, p k ) with respect to the 1:1 (alternatively 1+1) protection architecture. u Proof  Remove from the network all the links that are not used by the paths of (p 1,p 2,…, p k ). We have to show that there exists a pair of paths in the resulting network such that  Assign to each link two units of capacity, and assign to all other links one unit of capacity.  There exists a pair of paths that intersect only on links from iff it is possible to define an integral link flow that transfers two flow units from s to t.  Hence, it is sufficient to show that it is possible to define an integral link flow that transfers two flow units from s to t.

64. Two Paths are Enough u Proof (cont)  However, since all capacities are integral, the maximum flow that can be transferred from s to t is equal to the maximum flow that can be transferred from s to t when the integrality restriction is omitted. Hence, we left to show that it is possible to transfer two flow units from s to t.  Suppose by the way of contradiction that it is impossible to transfer two flow units from s to t.  Hence, according to the max-flow min cut theorem there exists a cut (S,T) with s  S and t  T such that  Therefore, since the capacity of all links is integral it follows that C(S,T)≤1.  Hence, since each link has at least one unit of capacity, it follows that at most one link crosses (S,T).  Denote this link by e. Since C(S,T)≤1 it follows that c e ≤1.  Obviously all paths from (p 1,p 2,…, p k ) must traverse through e. Hence, Therefore, by construction c e =2, which contradicts the fact that c e ≤1.

65. Establishing the widest p-survivable connection u Why is it enough to perform the search over the set  If one path admits a link e then the bandwidth of the connection is at most c e.  If both paths admit a link e then the bandwidth of the connection is at most c e /2.  Hence, by definition, there exists at least one tight link e  E such that the bandwidth of the connection is either c e or c e /2. u Why O(logN) executions of a min cost flow algorithm ?  The set contains 2·M elements.  A binary search over the set enables to consider O(log2·M)=O(logN) values.

66. The end-to-end delay restriction is intractable  A special case of our problem: Is there a path flow that transfers  flow units from s to t such that if path p transfers a positive amount of flow then D(p) ≤ D?  The partition problem: Given an ordered set of elements a 1, a 2,…, a 2n that constitute a set A with a size s(a)  + for each a  A, is there a subset A’  A such that A’ contains exactly one element of a 2i-1, a 2i for 1 ≤ i ≤ n such that ∑ a  A’ s(a)=∑ a  A\A’ s(a)? u All link capacities are 1.  Claim: It is possible to transfer 2 flow units over paths whose end-to-end delays are not larger than ½∑ a  A s(a) iff there is a subset A’  A such that A’ contains exactly one element of a 2i-1, a 2i for 1 ≤ i ≤ n and ∑ a  A’ s(a)=∑ a  A\A’ s(a). S(a 1 ) S(a 3 ) S(a 5 ) S(a 2n-1 ) S T S(a 2 )S(a 4 ) S(a 6 ) S(a 2n )

67. The end-to-end delay restriction is intractable <=  There is a a subset A’  A such that A’ contains exactly one element of a 2i-1, a 2i for 1≤i≤n and ∑ a  A’ s(a)=∑ a  A\A’ s(a).  The selection of the links that correspond to the elements of A’ and the zero delay links that connect these links constitutes a path p.  Path p is disjoint to the path that the complement subset A\A’ defines.  Since all capacities are equal to 1, we have two disjoint paths that can transfer together 2 units of flow.  The end-to-end delay of each path is ½∑ a  A s(a).=>  There is a path flow that transfers two flow units over paths that are not larger than ½∑ a  A s(a).  Let p be a path that carries a positive flow; by construction, p contains exactly one element of a 2i-1, a 2i for 1≤i≤n.  Since all the links have one unit of capacity p can transfer at most 1 flow unit.  Therefore, there exists a path p’ that is disjoint to p that transfers a positive flow; by construction, p’=A\p  Hence, D(p) ≤ ½∑ a  A s(a) and D(p’) ≤ ½∑ a  A s(a).  Therefore, since D(p)+ D(p’)=∑ a  A s(a) it follows that ∑ a  p s(a)=∑ a  p’ s(a)=½∑ a  A s(a).

68. The delay jitter restriction is intractable  A special case of our problem: Is there a path flow that transfers  flow units from s to t such that if path p 1, p 2 transfers a positive amount of flow then D(p 1 )-D(p 2 ) ≤ J? u Reduction from the problem with end-to-end delay restriction. S T A link with a capacity ∑c e and a zero delay. It is possible to transfer  flow units in network A over paths with end-to-end delay at most W iff it is possible to transfer  +∑ce flow units in network B over paths with delay jitter restriction W. S T A B

69. The restriction on the number of paths is intractable u A special case of our problem: Is there a path flow that transfers  flow units from s to t over at most K paths?  The single source unsplittable flow problem: Given a network G with a source s, targets t 1, t 2,…, t k and corresponding demands D 1, D 2,…, D k, is there an assignment of traffic to paths such that for each 1 ≤ i ≤ k demand D i is routed over a single path without violating the capacity constraints?  Claim: There exists a path flow that transfers  = D 1 + D 2 +…+ D k flow units from S to T over at most K paths iff it is possible to find an assignment of the demands D 1, D 2,…, D k to paths such that D i, 1 ≤ i ≤ k is routed over a single path without violating the capacity constraints  There is exactly one path from S to t i for each 1 ≤ i ≤ k. Hence, there are exactly K paths from S to T that carry a positive flows.  There is at least one path from S to t i for each 1 ≤ i ≤ k. However, since there are at most K paths there is exactly one path from S to t i for each 1 ≤ i ≤ k. S t1t1 t2t2 tktk T D1D1 D2D2 DkDk

70. Waxman and Power-law topologies u Waxman networks:  Source and destination are located at the diagonally opposite corner of a square area of unit dimension.  198 nodes are uniformly spread over the square.  A link between two nodes u,v exists with a probability, which depends on the distance between them δ(u,v): where α=1.8, β=0.05. u Power-law networks:  We assigned a number of out-degree credits to each node, using the power-law distribution β∙x -α where α=0.75 and β=0.05.  Then, we connected the nodes so that every node obtained the assigned out-degree.

71. Minimizing the congestion under delay-jitter restrictions

72. Approximation scheme for the restriction on the delay jitter u We impose a restriction 1≤H≤N-1 on the hop count.  Important in order to cope with routing loops. u We present an approximation scheme for the case where d max =O(J).  The number of variables is in the order of M∙H∙min{ D,H∙d max } ≤ M∙H 2 ·d max. u The delay of each link is reduced to smaller integral value. u Total error in the evaluation of the delay of each path is H∙Δ. u A pair of paths that originally have a delay jitter J may now have a delay jitter J+H∙Δ. u Therefore, in order to relax the new instance the delay jitter restriction is:

73. Approximation scheme for the restriction on the delay jitter u Assume that p 1, p 2 transfers a positive flow in the output. We will show that D(p 1 )-D(p 2 )≤J(1+є).

74. Approximation scheme for the restriction on the delay jitter u Assume that p transfers a positive flow in the output. We will show that D(p) ≤D(1+є).

75. Existence of Nash Equilibrium u The joint strategy space is finite.  Each user selects at most N out of |P (s,t) | possible paths.  There are at most |U| users. u By the way of contradiction assume that there is no Nash equilibrium. u Each profile in the joint strategy space has a player that can improve its bottleneck. u Let be a sequence of profiles such that for each two profiles f i, f i+1  exactly one user in f i+1 reroutes its traffic and improves its bottleneck with respect to f i. u After a finite number of transitions between successive profiles we must encounter the same profile. u Let u be a user that achieves the worst (not constant) bottleneck in all profiles.  Let f k be the profile where u achieves for the first time the worst bottleneck. u There exists in profile f k-1 exactly one user u’ that improves its bottleneck. u However, since u’ ships traffic through the bottleneck of u in f k, u’ is not improving its bottleneck.

76. No price of anarchy for bottleneck network objectives Theorem: Given an instance [G(V,E), U,q e (·)]. If multipath routing is allowed than the price of anarchy is 1. proof : u Notations  f- Nash flow.  (f)- The collection of users that ship traffic through a network bottleneck in f.  g- Path flow f without the users U\  (f) and their respective flows.  E’ – The collection of all network bottlenecks with respect to g.  P(e)- The collection of all paths that traverse through link e. u Lemma: g is a Nash flow that satisfies  B(f)=B(g)  b u (g)=B(g) for each user u   (f).  Proof.

77. No price of anarchy for bottleneck network objectives (cont.) u By contradiction assume the existence of a flow vector h, B(h)<B(g) u Since g is a Nash flow, every path p  P (s u,t u ) where u  (f) must traverse through at least one network bottleneck from E’.  Therefore, for each bottleneck u   (f). u Therefore, u Therefore, since the total traffic of every feasible flow vector that traverses through the paths equals to, the total traffic that traverse through equals to both in g and in h.

78. No price of anarchy for bottleneck network objectives (cont.) u Since B(h)<B(g) it follows that q e (h e )<q e (g e ) for each e  E’. u Therefore, h e < g e for each e  E’. u Therefore, the traffic that traverses through P(e) is smaller in h than in g for each e  E’. u Therefore, the traffic that traverses through is smaller in h than in g. u However, this contradicts the fact that the total traffic of the paths in is the same in flow vector h and g.  Since B(g) is optimal and since B(f)=B(g), it follows that B(f) is optimal (the bottleneck of f that also satisfy the demands of all the users in  (f) can only be worse than the bottleneck of g)

79. Proof of the Lemma u Let E’’ be the collection of all bottlenecks with respect to f. u B(f)=B(g):  By definition, the traffic that is carried over E’’ belongs only to  (f).  Therefore, since for each u  (f) and p  P, it holds that for each e  E’’.  Therefore, B(f)=B(g).  b u (g)=B(g) for each user u   (f).  Consider a user u   (f).  u must ship traffic through at least one link from E’’ in flow vector f.  Since for each u  (f) and p  P, it follows that u must also ship positive traffic through a link from E’’ in flow vector g.  Since q e (g e )=q e (f e )=B(f) for each e  E’’, it follows that b u (g)=B(g). u g is at Nash equilibrium:  Since f is a Nash flow, every path p  P (s u,t u ) where u  (f) must traverse through at least one network bottleneck from E’’.

80. Proof of the Lemma  We have shown that all bottlenecks of f remain unchanged in g.  Therefore, every path p  P (s u,t u ) where u  (f) traverses through one network bottleneck with respect to g.  By contradiction, assume there exists a user u  (f) in g, that can improve its bottleneck.  Let E (s u,t u ) be the collection of all network bottlenecks in g on paths from P (s u,t u ).  Let P(e) be the collection of all paths that traverse through e.  u can improve its bottleneck only if it reduces the total traffic that it carries over paths from P(e) for each employed link e  E (s u,t u ).  Therefore, it must ship traffic to other paths from P (s u,t u ).  However, we have shown that all other paths already traverse through at least one bottleneck from E (s u,t u ).

81. Minimizing congestion while restricting the number of paths u Theorem: The optimal network congestion factor of a  /K-integral path flow is larger by a factor of at most 2 than the optimal network congestion factor of a path flow that admits at most K r paths for each request r  R. u Proof: Let f * be a path flow that has the smallest network congestion factor α * among all path flows that transfers for each r  R,  r flow units from S r to T r over at most K r paths. f=2∙f* is a path flow with a network congestion factor 2∙α* that transfers 2  r flow units from S r to T r over at most K r paths for each r  R. For each r  R, round down the flow f(p) over each path p  P (s r,t r ) to a multiple of  r /K r. Let f R be the resulting path flow. Given a network G(V,E) and a source- destination pair. For each r  R, f transfers 2  r flow units over at most K r paths. Therefore, f R transfers at least  r flow units from S r to T r for each r  R f R is a  /K - integral path flow that transfers at least  r flow units from S r to T r for each r  R and has a network congestion factor of at most 2∙ α *.