Multipath Routing

1
Multipath Routing

• Ph.D. Research Proposal
• Ron Banner
• Supervisor Prof. Ariel Orda
• March 2004

2
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Selfish multipath routing
• Online multipath routing for congestion
  minimization
• Future research

3
What is Multipath Routing?

• Multipath Routing is the method of establishing
  multiple paths between given source-destination
  nodes within the network.

4
Advantages of Multipath Routing

• Survivability
• Provides redundancy.
• Congestion avoidance
• Improves network utilization.
• Provides load balancing.
• Management and control
• Provides better performance in the presence of
  selfish/unregulated behavior

5
Previous Research

• Survivability
• Mainly solutions that focus on the establishment
  of pairs of disjoint paths (e.g., the 11 and 11
  protection architectures).
• Congestion avoidance
• Mainly heuristics (e.g., ECMP).
• Online no previous work for multipath routing.
• 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
• de-delay of link e
• ce-capacity of link e
• pe-failure probability of link e
• fe-flow rate on link e

7
Summary of results Survivability

• We provide a quantitative framework that
  specifies the desired level of survivability
  against single failures.

c30, p0.05
c10, p0.05
c20, p0.05
c30, p0
c30, p0.05
S
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c30,p0.05
c30, p0.05
8
Summary of results Survivability

• We developed optimal polynomial schemes for 11
  and 11 protection that consider important
  tradeoffs
• Survivability vs. bandwidth
• Survivability vs. feasibility.
• No need to establish connections that consist of
  more than two paths.
• Derived a new hybrid protection architecture
  that has several advantages over both the 11 and
  11 protection architecture.
• Show that by just slightly alleviating the
  requirement of full survivability a major
  improvement is obtained.

9
Summary of resultsCongestion minimization-offlin
e

• Goal Minimize network congestion when all
  demands are known in advance.
• Cope with constraints
• Delay jitter
• End-to-end delay
• Number of paths
• Minimizing the congestion under end-to-end delay
  and/or delay jitter
• NP-hard
• Pseudo polynomial solution
• e-optimal approximation scheme
• Minimizing the congestion while restricting the
  number of routing paths
• NP-hard
• 2-approximation scheme

10
Summary of results Congestion
minimization-online

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

11
Summary of resultsSelfish multipath routing

• Goal Investigating the degradation in network
  performance due to selfish behavior of users.
• Given a load-dependent performance function
  qe(fe) for each link we consider bottleneck
  network objectives i.e., Maxe?Eqe(fe) and
  additive network objectives i.e.,
• Assume that users are selfish and their
  performance is dictated by their worst
  (bottleneck) elements.

12
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Selfish multipath routing
• Online multipath routing for congestion
  minimization
• Future research

13
The tunable survivability concept

• Current survivability schemes typically offer two
  degrees of protection against single failures
• Full (100) protection.
• No protection at all.
• 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).
• Tunable survivability enables to consider
  valuable tradeoffs.
• Survivability vs. bandwidth
• Survivability vs. feasibility
• Survivability vs. end-to-end delay

14
Survivable connections

• p-survivable connection a collection of paths
  (p1,p2,, pk)?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 (p1,p2,,
  pk) remains operational.

• The bandwidth of a survivable connection with
  respect to the 11 protection architecture is the
  maximum B0 such that nBce for each link e that
  is common to n paths from (p1,p2,, pk).

15
Two Paths are Enough

• Theorem Let (p1,p2,, pk)?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 (p1,p2,, pk) with
  respect to the 11 (alternatively 11) protection
  architecture.
• Proof (sketch for the 11 protection)
• We shall construct only from the links
  that belong to paths in (p1,p2,, pk).
  Therefore, the bandwidth of is at least
  that of (p1,p2,, pk).
• Formal proof

16
Most Survivable Connections with a Bandwidth of
at Least B

• Since two paths are enough, we focus on
  survivable connection that consist of two paths.
• The most survivable connection with a bandwidth
  of at least B for the 11 protection architecture
  is established by a reduction to the min cost
  flow problem.
• The flow demand is set to 2B flow units.

Links in the transformed network
Discard the link
CeltB
A link in the original network
ceB, we0
ce,pe
BCelt2B
Ce2B
ceB, we0
ceB, we-ln(1-pe)
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Most Survivable Connections with a Bandwidth of
at Least B

• Since the flow demand and capacities are
  B-integral the min cost flow is B-integral.
• The flow decomposition algorithm can be applied
  in order to decompose the B-integral link flow
  (that transfers 2B flow units) into a flow over
  two paths p1, p2 such that f(p1)f(p2)B.
• Since the flow has a minimum cost,
• has a minimum value.
• Therefore, (p1,p2 ) is a connection with a
  bandwidth of at least B that maximizes
  hence, it maximizes

18
Establishing Most and Widest p-survivable
Connections

• 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 B0.
• The widest p-survivable connection is the
  p-survivable connection with the maximum
  bandwidth.
• How to establish the widest p-survivable
  connection?
• Idea search for the largest B such that the most
  survivable connection with a bandwidth of at
  least B is a p-survivable connection.
• It is enough to perform a binary search over the
  set
• Why
• The widest p-survivable connection is therefore
  established within O(logN) executions of any min
  cost flow algorithm.
• Why

19
Establishing Survivable Connections for 11
protection

• The only difference in the reduction lies for the
  links that have capacities in the range B,2B.
• For 11 protection only one of the paths carries
  B flow units.
• Hence, all links that have a capacity in the
  range B,2B can concurrently be employed by both
  paths.

Links in the transformed network
A link in the original network
Discard the link
CeltB
ce,pe
CeB
ceB, we0
ceB, we-ln(1-pe)
Go to 11 reduction
20
The Hybrid protection architecture

• The tunable survivability concept gives rise to a
  third protection architecture.
• Reduces the congestion of all links that are
  shared by both paths w.r.t 11 protection.
• Upon a link has a faster restoration w.r.t 11
  protection.
• Provides the fastest propagation of data.
• However, requires additional nodal capabilities.

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T
21
The Hybrid protection architecture

• The hybrid architecture transfers through each
  link exactly one duplicate of the original
  traffic.
• Hence, the bandwidth of (p1,p2) with respect to
  hybrid protection is
• Hence, by definition, all schemes for 11
  protection apply for hybrid protection.

Go to Def
22
Simulation results

• We quantify how much we gain by employing tunable
  survivability instead of full survivability.
• Random networks
• 10,000 Waxman topologies
• 10,000 Power-law topologies.
• Explain the construction

Bandwidth ratio (11)
23
Simulation results

Feasibility ratio
Bandwidth ratio (11)
24
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Selfish multipath routing
• Online multipath routing for congestion
  minimization
• Future research

25
Problem formulation

• Goals
• Minimize network congestion when all demands are
  known in advance
• Cope with constraints (delay-jitter, delay,
  number of paths)
• 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

• Restricting the delay-jitter among all routing
  paths
• RFC 2991.
• Avoid the fast retransmit mode.
• Reduce buffering requirements.
• 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.
• Bounding the end-to-end delay of each path.

27
Computational Intractability

• Minimizing the network congestion factor under
  the end-to-end delay restriction is NP- hard.
• Proof .
• Minimizing the network congestion factor under
  the delay jitter restriction is NP- hard.
• Proof .
• 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

• 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.
• Proof

Let f be a path flow that has the smallest
network congestion factor a among all path flows
that transfers ? flow units from S to T over at
most K paths.
Given a network G(V,E) and a source-destination
pair.
f2f is a path flow with a network congestion
factor 2a 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 fR be the resulting path
flow.
fR 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 a.
Since f transfer 2? flow units over at most K
paths fR transfers at least ? flow units from S
to T
29
Minimizing the congestion under integrality
restrictions

• A ?/K-integral path flow admits at most K paths.
• Corollary minimizing the congestion while
  restricting the flow to be integral in ?/K is a
  2-approximation scheme.
• 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.
• multiply all link capacities by a factor of a.
• 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.
• 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

• It is straight forward to extend the linear
  program to the multi-commodity case.
• The path flow is constructed using a variant of
  the flow decomposition algorithm.
• 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

• Goal reduce the value of the end-to-end delay
  restriction D.
• Delete from the network all the links with a
  delay degtD.
• Delay scaling
• Apply the linear program for the new instance.
• As the new instance relax the original instance
  the congestion is not worse then the optimum.
• Convert each non-simple path into a simple path.
• Total error for a path N??.
• New end-to-end delay D N??D(1?).

33
Minimizing the congestion under delay-jitter
restrictions

• Idea restrict the minimum end-to-end delay L and
  the maximum end-to-end delay U of the routing
  paths.
• It is sufficient to add the linear program a
  minimum end-to-end delay restriction L.
• New Linear Program .
• 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, LJ, respectively.
• Scaling down the end-to-end delay restriction D
  produces an ?-optimal approximation scheme for
  the case where dmaxO(J).
• Details .

34
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Selfish multipath routing
• Online multipath routing for congestion
  minimization
• Future research

35
Selfish Routing

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

36
Previous Work

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

37
Model

• A set of users U.
• For each user, a positive flow demand ?u and a
  source-destination pair (su,tu).
• For each link e, a performance function qe().
• qe() is continuous and increasing for all links.
• Users behavior
• Users are selfish.
• They optimize bottleneck objectives
• Network
• Bottleneck objective
• Additive objective

38
Non-uniqueness of Nash Equilibrium

• One user wants to transfer 1 unit from s to t.
• Assume that qe(fe)fe for each e?E.
• (fp11, fp20) (fp10, fp21) are Nash flows
  with respect to unsplittable flow vectors.
• (fp10.5, fp20.5) (fp10.25, fp20.75) are
  Nash flows with respect to splittable flow
  vectors.
• We identified two different Nash flow for each
  routing approach.

p1
e1
e3
s
t
e2
p2
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.
• 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 N1.
• The existence of NEP for Multipath Routing
  corresponds to the case where N?8.
• However, still needs to prove for the case where
  N8.
• The proof of the theorem .

40
No price of anarchy for bottleneck network
objectives

• 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
• Theorem Given an instance G(V,E), U,qe(?). If
  multipath routing is allowed then the price of
  anarchy is 1.
• Proof .
• Braess paradox the addition of links to
  noncooperative networks can negatively impact
  performance of all users.
• However, cannot occur for multipath routing (when
  qe(0)0).

41
Price of anarchy is at most M with additive
objectives

• Theorem Given an instance G(V,E), U,qe(?). If
  multipath routing is allowed than the price of
  anarchy with respect to additive network
  objectives is M.
• Proof
• Let f and f denote a Nash and an optimal flow,
  correspondingly.
• Therefore, B(f)B(f).
• Therefore, maxe?E qe(f) maxe?E qe(f).
• Hence, ?e?E qe(f) MmaxEqe(f) Mmaxe?E
  qe(f) M?e?E qe(f)
• 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

• The price of anarchy is unbounded for single path
  routing
• Additive network objectives.
• Bottleneck network objectives.

?A ?
?B 2?
Optimal flow
Nash flow
Price of anarchy
Bottleneck
Additive
43
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Selfish multipath routing
• Online multipath routing for congestion
  minimization
• Future research

44
The Model

• Requests arrive one at a time and there is no a
  priori knowledge regarding future demands.
• Each request specifies
• the source sr and destination tr.
• the requested flow demand ?r.
• the maximum number of routing paths kr that can
  carry the demand.
• Goal Route all demands while minimizing the
  network congestion factor .
• 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

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

46
Minimizing the congestion under integrality
restrictions

• A path flow is ?/K-integral if the flow of each
  request r?R over each path is integral in ?r/Kr.
• 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 Kr
  paths for each request r?R .
• Proof .
• A ?/K-integral path flow employs at most Kr paths
  for each r?R.

• Corollary minimizing the congestion while
  restricting the flow to be integral in ?/K is a
  2-approximation scheme.

47
Online solution

• Upon the arrival of the nth request
• Split the request to Kn successive requests to
  transfer ?n/Kn flow units.
• Employ the online strategy of plotkin at el to
  route the demands over single paths.
• Plotkins online strategy produces a competitive
  ratio of O(logN).
• Therefore, we establish an online strategy with a
  competitive ratio of O(logN) for ?/K-integral
  path flows.
• Therefore, we establish an online strategy for
  our original problem with a competitive ratio of
  2?O(logN)O(logN).

48
A Lower Bound of O(logN) for Multipath Routing
V1
V2
V3
S
VN-1
VN
49
A Lower Bound of O(logN) for Multipath Routing
(cont.)

• After logN requests the network congestion factor
  is at least ½logN.
• The optimal offline algorithm can achieve a
  network congestion factor of 1.

V1
V2
V3
S
VN-1
VN
50
A Lower Bound of O(logN) for Multipath Routing
(cont.)

• There exists a lower bound of ½logN for
  networks with at most NNlogNN2NlogN nodes.
• We have to show that ½logN?(logN).
• Indeed, there exists Cgt0 and NgtN0 such that
  logNlogNlog(2logN)logNlog2loglogN C
  ½logN.

• There exists a lower bound of ?(logN) for the
• best possible competitive ratio.

Our online algorithm is best possible.
51
Agenda

• Introduction summary of results
• Multipath routing schemes for survivable networks
• Multipath routing schemes for congestion
  minimization
• Online multipath routing for congestion
  minimization
• Selfish multipath routing
• Future research

52
Future research

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

53
Deepening the Current Work

• 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.
• 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

• In networks that have many users, the price of
  anarchy with respect to additive metrics may be
  very large.
• 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.
• Driving users to route traffic according to
  bottleneck metrics bounds the price of anarchy to
  M.
• 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

• We have established an online strategy for
  permanent connections (i.e., connections with
  infinite holding times).
• In practice, the holding times are usually
  finite.
• 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.
• Investigate multipath routing for the finite
  holding time model.
• Investigate the lower bound.
• Establish corresponding multipath routing
  schemes.

56
Other Congestion Criteria

• Thus far, we measured congestion according to the
  most utilized links in the network.
• Although these links are the most severely
  affected by congestion, other links are affected
  as well.
• Moreover, there are cases where congestion is
  better modeled through non-linear optimization
  functions.
• 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

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

58
Recovery Schemes for Multipath Routing

• Multipath Routing has the advantage of fast
  restoration upon a failure.
• 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.
• Requires that the sum of the spare capacities of
  the remaining paths is not smaller than the flow
  on the failed path.
• Establish multipath routing schemes that enable
  fast recovery while considering the requirements
  of multipath routing.

59
Multipath Routing and Wireless networks

• 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 networks
  lifetime while considering the requirements of
  multipath routing.
• 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

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

61
Time Dependent Flow Demands in Multipath Routing

• We have assumed that flow demands are constant in
  time.
• 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

• Theorem Let (p1,p2,, pk)?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 (p1,p2,, pk) with
  respect to the 11 (alternatively 11) protection
  architecture.
• Proof
• Remove from the network all the links that are
  not used by the paths of (p1,p2,, pk). 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

• 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 ce1.
• Obviously all paths from (p1,p2,, pk) must
  traverse through e. Hence, Therefore, by
  construction ce2, which contradicts the fact
  that ce1.

65
Establishing the widest p-survivable connection

• 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 ce.
• If both paths admit a link e then the bandwidth
  of the connection is at most ce/2.
• Hence, by definition, there exists at least one
  tight link e?E such that the bandwidth of the
  connection is either ce or ce/2.
• Why O(logN) executions of a min cost flow
  algorithm ?
• The set contains 2M elements.
• A binary search over the set enables to consider
  O(log2M)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 a1, a2 ,, a2n 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 a2i-1, a2i for 1in such that
  ?a?A s(a)?a?A\A s(a)?
• 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 a2i-1,
  a2i for 1in and ?a?A s(a)?a?A\A s(a).

67
The end-to-end delay restriction is intractable

• lt
• There is a a subset A?A such that A contains
  exactly one element of a2i-1, a2i for 1in 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).
• gt
• 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
  a2i-1, a2i for 1in.
• 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, pA\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 p1, p2 transfers a positive amount
  of flow then D(p1)-D(p2)J?
• Reduction from the problem with end-to-end delay
  restriction.

A link with a capacity ?ce and a zero delay.
S
T
B
A
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.
69
The restriction on the number of paths is
intractable

• 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 t1, t2
  ,, tk and corresponding demands D1, D2 ,, Dk ,
  is there an assignment of traffic to paths such
  that for each 1ik demand Di is routed over a
  single path without violating the capacity
  constraints?
• Claim There exists a path flow that transfers ?
  D1 D2 Dk flow units from S to T over at
  most K paths iff it is possible to find an
  assignment of the demands D1, D2 ,, Dk to paths
  such that Di, 1ik is routed over a single path
  without violating the capacity constraints
• There is exactly one path from S to ti for each
  1ik. Hence, there are exactly K paths from S to
  T that carry a positive flows.
• There is at least one path from S to ti for each
  1ik. However, since there are at most K paths
  there is exactly one path from S to ti for each
  1ik.

S
tk
t1
t2
D2
Dk
D1
T
70
Waxman and Power-law topologies

• 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 d(u,v)
• where a1.8, ß0.05.
• Power-law networks
• We assigned a number of out-degree credits to
  each node, using the power-law distribution ßx-a
  where a0.75 and ß0.05.
• 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

• We impose a restriction 1HN-1 on the hop count.
• Important in order to cope with routing loops.
• We present an approximation scheme for the case
  where dmaxO(J).
• The number of variables is in the order of
  MHmin D,Hdmax MH2?dmax.
• The delay of each link is reduced to smaller
  integral value.
• Total error in the evaluation of the delay of
  each path is H?.
• A pair of paths that originally have a delay
  jitter J may now have a delay jitter JH?.
• Therefore, in order to relax the new instance the
  delay jitter restriction is

73
Approximation scheme for the restriction on the
delay jitter

• Assume that p1, p2 transfers a positive flow in
  the output. We will show that D(p1)-D(p2)J(1?).

74
Approximation scheme for the restriction on the
delay jitter

• Assume that p transfers a positive flow in the
  output. We will show that D(p) D(1?).

75
Existence of Nash Equilibrium

• 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.
• By the way of contradiction assume that there is
  no Nash equilibrium.
• Each profile in the joint strategy space has a
  player that can improve its bottleneck.
• Let ltf1,f2, gt be a sequence of profiles such
  that for each two profiles fi, fi1?ltf1,f2, gt
  exactly one user in fi1 reroutes its traffic and
  improves its bottleneck with respect to fi.
• After a finite number of transitions between
  successive profiles we must encounter the same
  profile.
• Let u be a user that achieves the worst (not
  constant) bottleneck in all profiles ltf1,f2,fn
  gt.
• Let fk be the profile where u achieves for the
  first time the worst bottleneck.
• There exists in profile fk-1 exactly one user u
  that improves its bottleneck.
• However, since u ships traffic through the
  bottleneck of u in fk, u is not improving its
  bottleneck.

76
No price of anarchy for bottleneck network
objectives

• Theorem Given an instance G(V,E), U,qe(?). If
  multipath routing is
• allowed than the price of anarchy is 1.
• proof
• Notations
• f- Nash flow.
• G(f)- The collection of users that ship traffic
  through a network bottleneck in f.
• g- Path flow f without the users U\G(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.
• Lemma g is a Nash flow that satisfies
• B(f)B(g)
• bu(g)B(g) for each user u?G(f).
• Proof .

77
No price of anarchy for bottleneck network
objectives (cont.)

• By contradiction assume the existence of a flow
  vector h, B(h)ltB(g)
• Since g is a Nash flow, every path p?P(su,tu)
  where u??(f) must traverse through at least one
  network bottleneck from E.
• Therefore, for each
  bottleneck u?G(f).
• Therefore,
• 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.)

• Since B(h)ltB(g) it follows that qe(he)ltqe(ge) for
  each e?E.
• Therefore, helt ge for each e?E.
• Therefore, the traffic that traverses through
  P(e) is smaller in h than in g for each e?E.
• Therefore, the traffic that traverses through
  is smaller in h than in g.
• 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
  G(f) can only be worse than the bottleneck of g)

79
Proof of the Lemma

• Let E be the collection of all bottlenecks with
  respect to f.
• 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).
• bu(g)B(g) for each user u?G(f).
• Consider a user u?G(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 qe(ge)qe(fe)B(f) for each e ?E, it
  follows that bu(g)B(g).
• g is at Nash equilibrium
• Since f is a Nash flow, every path p?P(su,tu)
  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(su,tu) 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(su,tu) be the collection of all network
  bottlenecks in g on paths from P(su,tu).
• 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(su,tu).
• Therefore, it must ship traffic to other paths
  from P(su,tu).
• However, we have shown that all other paths
  already traverse through at least one bottleneck
  from E(su,tu).

81
Minimizing congestion while restricting the
number of paths

• 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 Kr
  paths for each request r?R .
• Proof

Let f be a path flow that has the smallest
network congestion factor a among all path flows
that transfers for each r?R, ?r flow units from
Sr to Tr over at most Kr paths.
Given a network G(V,E) and a source-destination
pair.
For each r?R, round down the flow f(p) over each
path p?P(sr,tr) to a multiple of ?r/Kr. Let fR be
the resulting path flow.
f2f is a path flow with a network congestion
factor 2a that transfers 2?r flow units from Sr
to Tr over at most Kr paths for each r?R.
fR is a ?/K - integral path flow that transfers
at least ?r flow units from Sr to Tr for each r?R
and has a network congestion factor of at most 2
a.
For each r?R, f transfers 2?r flow units over at
most Kr paths. Therefore, fR transfers at least
?r flow units from Sr to Tr for each r?R