IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 2, APRIL 2007 413
Multipath Routing Algorithms for
Congestion Minimization
Ron Banner, Senior Member, IEEE, and Ariel Orda, Fellow, IEEE
Abstract—Unlike traditional routing schemes that route all
traffic along a single path, multipath routing strategies split the
traffic among several paths in order to ease congestion. It has been
widely recognized that multipath routing can be fundamentally
more efficient than the traditional approach of routing along single
paths. Yet, in contrast to the single-path routing approach, most
studies in the context of multipath routing focused on heuristic
methods. We demonstrate the significant advantage of optimal
(or near optimal) solutions. Hence, we investigate multipath
routing adopting a rigorous (theoretical) approach. We formalize
problems that incorporate two major requirements of multipath
routing. Then, we establish the intractability of these problems in
terms of computational complexity. Finally, we establish efficient
solutions with proven performance guarantees.
Index Terms—Computer networks, congestion avoidance,
routing protocols.
I. INTRODUCTION
C
URRENT routing schemes typically focus on discovering
a single “optimal” path for routing, according to some de-
sired metric. Accordingly, traffic is always routed over a single
path, which often results in substantial waste of network re-
sources.
Multipath routing is an alternative approach that dis-
tributes the traffic among several “good” paths instead of routing
all traffic along a single “best” path.
Multipath routing can be fundamentally more efficient than
the currently used single-path routing protocols. It can signifi-
cantly reduce congestion in “hot spots,” by deviating traffic to
unused network resources, thus, improving network utilization
and providing load balancing [16]. Moreover, congested links
usually result in poor performance and high variance. For such
circumstances, multipath routing can offer steady and smooth
data streams [6].
Multipath routing algorithms that optimally split traffic be-
tween a given set of paths have been investigated in the con-
text of flow control (e.g., [14], [19], [20]). Yet, the selection
of the routing paths is another major design consideration that
has a drastic effect on the resulting performance. Therefore, al-
though many flow-control algorithms are optimal for a given set
of routing paths, their performance can significantly differ for
different sets of paths. Accordingly, in this paper, we focus on
multipath routing algorithms that both select the routing paths
and split traffic among them.
Manuscript received August 2, 2005; revised February 22, 2006; approved by
IEEE/ACM T
RANSACTIONS ON NETWORKING Editor R. Srikant. The conference
version of this paper appeared in Proceedings of the IFIP Networking, 2005.
The authors are with the Department of Electrical Engineering, Technion-
Israel Institute of Technology, Haifa 32000, Israel (e-mail: banner@tx.tech-
nion.ac.il; ariel@ee.technion.ac.il).
Digital Object Identifier 10.1109/TNET.2007.892850
Previous studies and proposals on multipath routing in
the previous context have focused on heuristic methods.In
[22], a multipath routing scheme, termed equal cost multipath
(ECMP), has been proposed for balancing the load along mul-
tiple shortest paths using a simple round-robin distribution. By
limiting itself to shortest paths, ECMP considerably reduces
the load balancing capabilities of multipath routing; moreover,
the equal partition of flows along the (shortest) paths (resulting
from the round robin distribution) further limits the ability to
decrease congestion through load balancing. OSPF-OMP [28]
allows splitting traffic among paths unevenly; however, the
traffic distribution mechanism is based on a heuristic scheme
that often results in an inefficient flow distribution. Both [29]
and [31] considered multipath routing as an optimization
problem with an objective function that minimizes the con-
gestion of the most utilized link in the network; however, they
focused on heuristics and did not consider the quality of the se-
lected paths. In [23], a scheme was presented to proportionally
split traffic among several “widest” paths that are disjoint with
respect to the bottleneck links. However, here too, the scheme
is heuristic and evaluated by way of simulations.
Through comprehensive simulations, we show that multipath
solutions obtained by optimal congestion reduction schemes are
fundamentally more efficient than solutions obtained by heuris-
tics. Specifically, we show that if the traffic distribution mech-
anism of the ECMP or OSPF-OMP schemes had been optimal,
the network congestion would have decreased by a factor of
more than 2.5; moreover, these simulations indicate that optimal
traffic distribution mechanisms become significantly more effi-
cient by just slightly alleviating the requirement to route along
shortest paths. Hence, the full potential of multipath routing is
far from having been exploited.
Accordingly, in this study, we investigate multipath routing
adopting a rigorous approach, and formulate it as an opti-
mization problem of minimizing network congestion. Under
this framework, we consider two fundamental requirements.
First, each of the chosen paths should usually be of satisfactory
“quality.” Indeed, while better load balancing is achieved by
allowing the employment of paths other than shortest, paths
that are substantially inferior (i.e., “longer”) may be prohibited.
Therefore, we consider the problem of congestion minimiza-
tion through multipath routing subject to a restriction on the
“quality” (i.e., length) of the chosen paths.
Another practical restriction is on the number of routing paths
per destination, which is due to the following reasons [23].
First, establishing, maintaining, and tearing down paths pose
considerable overhead. Second, the complexity of a scheme that
distributes traffic among multiple paths considerably increases
with the number of paths. Third, often there is a limit on the
1063-6692/$25.00 © 2007 IEEE
414 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 2, APRIL 2007
number of explicitly routing paths (such as label-switched paths
in MPLS [26]) that can be set up between a pair of nodes. There-
fore, in practice, it is desirable to use as few paths as possible
while at the same time minimize the network congestion.
Link state protocols [15] constitute an important class of
routing protocols that can be used for the implementation
of multipath solutions. In this class of protocols, each node
maintains a “map” of the network that enables it to compute the
routes. When a network link changes status, a notification is
flooded throughout the network and all nodes recompute their
routes according to their updated maps. It has been noted [15]
that link state protocols have the following desirable properties:
1) they can employ more complicated routing approaches and
can compute more accurate routes than can be computed with
distance vector protocols; 2) they respond faster to changes
and impose less communication overhead than the alternative
(distance vector) protocols; and 3) they are applicable for the
Internet (e.g., both OSPF and IS-IS are link state protocols)
and may be applicable for ad hoc networks (see, e.g., [25]).
Accordingly, in this paper, we assume a link-state routing
environment and formulate our problems accordingly.
Our Results: Consider first the problem of minimizing the
congestion under the requirement to route traffic along paths of
“satisfactory” quality. We first show that the considered problem
is NP-hard, yet admits a pseudo-polynomial solution. Accord-
ingly, we design two algorithms. The first is an optimal algo-
rithm with a pseudo-polynomial running time and the second
approximates the optimal solution to any desired degree of pre-
cision at the (proportional) cost of increasing its running time
(i.e., an
-optimal approximation scheme). In addition, we show
that these algorithms can be extended to offer solutions to reli-
ability related problems. Consider now the requirement of lim-
iting the number of paths per destination. We show that mini-
mizing the congestion under this restriction is NP-hard as well.
However, we establish a computationally efficient two-approx-
imation scheme for the problem, i.e., our algorithm provides a
solution that, in terms of congestion, is within a factor of at most
two away from the optimum.
Organization: In Section II, we introduce some terminology
and definitions, and formulate the main problems considered
in this study. In Section III, we consider the problem of min-
imizing congestion under path quality constraints and provide
both accurate as well as approximate solutions. In Section IV,
we investigate the problem of minimizing congestion subject to
a restriction on the number of paths per destination; we show
that the problem is NP-hard and provide a computationally effi-
cient two-approximation scheme. In Section V, we present sim-
ulation results that demonstrate the major advantage of optimal
congestion reduction schemes over two well-known heuristics.
Finally, Section VI summarizes the main results and discusses
future directions for future research.
II. M
ODEL AND PROBLEM FORMULATION
A network is represented by a connected directed graph
, where is the set of nodes and is the set
of links. Let
and .Apath is a finite
sequence of nodes
, such that, for 0
. A path is simple if all its nodes
are distinct. Given a source node
and a target node
is the set of (all) directed paths in from
to . Let represent the set of (all) simple paths
in
from to . Finally, for each path and
link
,define a link-path indicator , which is 1 if link
is contained in , and is 0 otherwise.
We consider a link-state routing environment, where each
source node has an image of the entire network. Each link
is assigned a length and a capacity . Given a
(nonempty) path
, the length of is defined as the sum of
lengths of its links, namely,
.
We consider two types of network flow representations. In
the path flow representation, each variable
is the flow on
some simple path
. Given two nodes and
a(flow) demand
, we say that a path flow is feasible iff it
satisfies the flow demand requirement, i.e.,
and the capacity constraints, i.e.,
for each . In the link flow representation, each variable
is the flow on some link . Given two nodes
and a demand , a link flow is feasible iff there
exists some feasible path flow
for the given instance such that
for each .
We proceed to formulate the criterion for congestion. Given
a network
and a link flow , the value is the
link congestion factor and the value
is the net-
work congestion factor. As noted in [2], [16], and [29], the net-
work congestion factor provides a good indication of conges-
tion. In [4], we show that the problem of minimizing the net-
work congestion factor is equivalent to the well-known max-
imum flow problem [1]. Hence, when there are no restrictions on
the paths (in terms of the number of paths or the length of each
path), one can find a path flow that minimizes the network con-
gestion factor in polynomial time through a standard max-flow
algorithm.
We are ready to formulate the two problems considered in
this study. The first problem aims at minimizing the network
congestion factor subject to a restriction on the “quality” (i.e.,
length) of each of the chosen paths.
Problem RMP (Restricted Multipath): Consider a network
, two nodes , a length , and a capacity
for each link , a demand , and a length
restriction
for each routing path. Find a feasible path flow
that minimizes the network congestion factor such that, if
is the set of paths in that are assigned a positive
flow, then, for each
, it holds that .
Remark 1: For convenience, and without loss of generality,
we assume that the length
of each link is not larger
than the length restriction
. Clearly, links that are longer than
can be erased.
The next problem considers the requirement to limit the
number of different paths over which a given demand is
shipped while at the same time minimizing the network con-
gestion factor.
Problem KPR (K-Path Routing): Given are a network
, two nodes , a capacity for each link
BANNER AND ORDA: MULTIPATH ROUTING ALGORITHMS FOR CONGESTION MINIMIZATION 415
, a demand , and a restriction on the number of
routing paths
. Find a feasible path flow that minimizes the
network congestion factor, such that, if
is the set of
paths in
that are assigned a positive flow, then .
Remark 2: In both problems, the source destination pair
is assumed to be connected, i.e., .
Remark 3: In both problem formulations, it is possible to
limit the link congestion factor
of each to any desired
congestion level
by replacing the given capacity value
with a new capacity value . Clearly, the capacity constraint
(that both problems must satisfy) assures that the
link congestion factor would be at most
.
III. M
INIMIZING
CONGESTION
UNDER PAT H
QUALITY
CONSTRAINTS
In this section, we investigate Problem RMP, i.e., the problem
of minimizing congestion under path quality constraints. In
Section III-A, we prove that Problem RMP is computational
intractable. Accordingly, in Section III-B, we establish a
pseudo-polynomial solution and in Section III-C we design an
-optimal approximation scheme for the problem.
A. Intractability of Problem RMP
We show that Problem RMP can be reduced to the Partition
Problem [12].
Theorem 1–Problem RMP is NP-hard:
Proof: Consider the following instance of the Partition
problem; given a set of elements
that constitute
a set
with size for each , find a subset
such that contains exactly one element of
for every and .
We transform Partition to RMP as follows (see also Fig. 1).
1) Given an element
with size ,define a unit
capacity link
with length .
2) For each link
define a link
and a link . Assign to both
a unit capacity and a zero length.
3) For each link
define a link
and a link . Assign to both a
unit capacity and a zero length.
4) Define links
and links
. Assign to each a unit capacity and a zero length.
5) Set:
and .
We shall prove that it is possible to transfer two flow units
over paths whose lengths are not larger than
without ex-
ceeding a network congestion factor of
iff there is a subset
such that contains exactly one element of
for every and .
(Remark: We refer to elements and their sizes interchange-
ably.)
Suppose there exists a subset such that
contains exactly one of for each and
. Then, it is easy to see that the
selection of the links that represents the elements in
and the
zero length links that connect those links constitutes a path.
Also, it is easy to see that this path is disjoint to the path that
the complement subset
defines. Since all capacities are
equal to 1, we have two disjoint paths that can transfer together
exactly two units of flow without violating the congestion
Fig. 1. Reduction of partition to RMP.
constraint . The length restriction is preserved since the
two defined paths have length of
, which was
defined to be the length restriction
. Suppose there is a
path flow that transfers two flow units over paths that are not
longer than
. Select one path that transfers a positive flow
and denote it as
.Define an empty set . For every link in ,
with length
, insert the element into . Since all links
in the graph have one unit of capacity, the selected path
is
not able to transfer more than one unit of flow. Now, delete
all the links that constitute path
. Since transfers at most
one unit of flow, there must be another path that is disjoint to
the selected path and transfers a positive flow over the links
that were left in the graph. For each link in that path with
size
, insert the element into a different set .We
will now prove that
, and, finally,
.
Since
and were constructed out of disjoint paths, it is
obvious that
. Since every path must traverse either
or for each and since both paths are
416 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 2, APRIL 2007
Fig. 2. Program RMP.
disjoint, . Finally, since both paths have
lengths that are not longer than
,wehave
(1)
Since
and , we get
(2)
Note that if variables
satisfy and in
addition
, it follows that .
Accordingly, we conclude from (1) and (2) that
.
Thus, problem RMP is NP hard.
B. Pseudo-Polynomial Algorithm for Problem RMP
The first step towards obtaining a solution to Problem RMP
is to define it as a linear program. To that end, we need some
additional notation.
Recall that we are given a network
, two nodes
, a length , and a capacity for each link ,
a demand
and a length restriction for each routing
path. Let
be the network congestion factor. Denote by the
total flow along
that has been routed from
to through paths with a total length . Finally, for each
, denote by the set of links that emanate from , and
by
the set of links that enter that node, namely,
and . Then,
Problem RMP can be formulated as a linear program over the
variables
as specified in Fig. 2.
The objective function (1) minimizes the network congestion
factor. Constraints (2), (3), and (4) are nodal flow conservation
constraints. Equation (2) states that the traffic flowing out of
node
, which has traversed through paths of length
, has to be equal to the traffic flowing into node ,
through paths
and links , such
that
; since , the length restriction is
obeyed; finally, (2) must be satisfied for each node other than
the source
and the target . Equation (3) extends the validity
Fig. 3. Single link flow can be decomposed into several path flows. Some of
them satisfy the length restriction and the rest violate it.
of (2) to hold for traffic that encounters source
after it has al-
ready passed through paths with non-zero length. Informally, (3)
states that “old” traffic that emanates from
not for the first time
(through a directed cycle that contains the source
) must sat-
isfy the nodal flow conservation constraint of (2), which solely
focuses on nodes from
. Equation (4) states that the
total traffic flowing out of source
, which has traversed paths
of length
, must be equal to the demand . Informally, (4)
states that the total “new” traffic that emanates from the source
for the first time must satisfy the flow demand . Equation (5)
is the link capacity utilization constraint. It states that the max-
imum link utilization is not larger than the value of the variable
, i.e., the network congestion factor is at most . Expression
(6) rules out nonfeasible flows and Expressions (7) and (8) re-
strict all variables to be nonnegative.
We can solve Program RMP (Fig. 1) using any polynomial
time algorithm for linear programming [18]. The solution to the
problem is then achieved by decomposing the output of Pro-
gram RMP (i.e., link flow
into a path flow that satisfies
the length restriction
. Standard flow techniques that transform
flows along links into flows along paths (e.g., the flow decompo-
sition algorithm [1]), cannot be used for our purpose since they
do not respect the length restrictions. This is illustrated by the
following example.
Example 1: Consider the network depicted in Fig. 3. Sup-
pose that the flow along each link is equal to one unit, i.e.,
for each . Moreover, assume that the length
restriction is 4. There are several path flows that can be decom-
posed out of the link flow
. For example, one such is the
path flow
that assigns one unit of flow to each of the paths
; indeed, since transfers one unit
of flow along each link, its link flow representation is
.
However, since the length of the path
is 6, the path
flow
violates the length restriction. On the other hand, if we
decompose the link flow
into the path flow that assigns one
unit to each of the paths
, the length
restriction is satisfied on all paths.
Our goal is, therefore, to establish an efficient algorithm
that decomposes link flow
into a path flow that satisfies
the length restrictions. Accordingly, consider Algorithm PFC,
which is specified in Fig. 4. This algorithm is an iterative algo-
rithm that identifies at each iteration a single path with a length
of at most
whose corresponding link flows are all posi-
tive; the path is identified through procedure path construction,
which is specified in Fig. 5. The flow over the corresponding
path is defined to be equal to the smallest flow
that belongs
to the path. Then, the algorithm subtracts the flow that traverses
through that path from the demand
and from each variable
in the path. The (iterative) algorithm repeats this process until
BANNER AND ORDA: MULTIPATH ROUTING ALGORITHMS FOR CONGESTION MINIMIZATION 417
Fig. 4. Algorithm PFC.
Fig. 5. Procedure path construction.
the demand is zeroed. Thus, the resulting path flow transfers
flow units along paths with length of at most . Finally, since
procedure path construction might return nonsimple paths, the
algorithm converts all nonsimple paths in the resultant path
flow into simple paths by eliminating their loops.
We turn to explain the main idea behind procedure
path construction (Fig. 5). The procedure identifies a path
, whose cor-
responding variables
are all
positive. Consider the positive variable
. Since
(6) of Program RMP zeroes each variable
with a length
, it holds for the positive variable
that ; hence, .
In other words, each path
with a corresponding
sequence of positive variables
has a length . Thus, in order
to establish a path
with a length of at most
, it is sufficient to find a sequence of positive variables
such that the link emanates
from source
and the link enters into the destination
, i.e., and . To that end, we employ
the following property that characterizes the solutions of
Program RMP. If a positive flow
enters through the
link
into the node , it follows that
there is some link
such that . Therefore,
since
for some link ; that emanates from the
source
, it is possible to follow positive flows (variables)
from
in order to construct a sequence of positive variables
such that .We
now prove that, by following these positive variables for a
finite number of times, we eventually encounter a positive
variable
such that enters the destination , i.e.,
we eventually identify a directed path from
to with a length
of at most
.
Lemma 1: Consider the nodes
identified by procedure
path construction (step 1). There exists an
, such that
the sequence
is a path from to .
Proof: It follows from constraint (4) that, since
,
there exists some link
such that the variable
is positive. Then, from constraints (2) and (3), it follows that, if
, then there exists some link such that the
variable
is positive. Thus, applying constraints (2) and (3)
for any index
, it follows that, if there exists a positive variable
, where , then, unless , there exists
a link
such that the variable is
positive. Therefore, if
for each , it must hold
that there exists an index
, such that . Hence, in
order to establish the lemma, it is sufficient to show that
for each .
Indeed, since
for each , it follows that
for any index ; hence, for each . There-
fore, since it follows from constraint (6) that
for each
, it holds that for each .
For completion, in Fig. 6, we specify Algorithm RMP, which
solves Problem RMP.
Next, we consider the complexity of Algorithm RMP. First,
it follows from [18] that the complexity incurred by solving the
linear program of step 1 is polynomial both in the number of
variables
and in the number of constraints needed to for-
mulate Program RMP. Thus, since both of these numbers are
in the order of
, the complexity of step 1 is polynomial
in
. Consider now the complexity incurred by step 2
(Algorithm PFC). Since, by construction, each iteration of Al-
gorithm PFC zeroes at least one variable
, it follows that Al-
gorithm PFC iterates for no more than the number of variables
. Moreover, since the complexity of each iteration is domi-
nated by the complexity of procedure path construction, which,
according to Lemma 1, consumes
operations, the com-
plexity of Algorithm PFC is
. Thus,
we conclude that the overall complexity of Algorithm RMP is
polynomial in
, i.e., Algorithm RMP is a pseudo-poly-
nomial time algorithm [12]. Whenever the value of
is poly-
nomial in the size of the problem, Algorithm RMP is a poly-
