An origin-based model for unique shortest path routing
Changyong Zhang
*
Department of Finance and Banking, Faculty of Business, Curtin University Sarawak, Miri, Malaysia
Link weights are the main parameters of shortest path routing protocols, the most commonly used protocols for IP
networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the
complexity of the constraints involved, there exist challenges to formulate the problem in such a way based on
which a more efficient solution algorithm than the existing ones may be developed. In this paper, an exact
formulation is first introduced and then mathematically proved correct. It is further illustrated that the formulation
has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition
method to solve the problem.
Journal of the Operational Research Society (2017) 68(8), 935–951. doi:10.1057/s41274-016-0144-9;
published online 16 December 2016
Keywords: mathematical modeling; model validation; constraint structure; decomposition; shortest path routing; link
weights
1. Introduction
Shortest path routing protocols such as OSPF are the most
widely deployed and commonly used protocols for IP
networks (Black, 2000; Moy, 1998; Tanenbaum and Wether-
all, 2011). They also find applications in, for example, road
networks (Abraham et al, 2010; Zhan and Noon, 1998). In
shortest path routing, each link is assigned a weight and traffic
demands are routed through the shortest paths with respect to
link weights (Bertsekas and Gallager, 1992), given by a
shortest path first algorithm (Bellman, 1958; Dijkstra, 1959;
Ford and Fulkerson, 2010). Link weights are hence the key
parameters, and an essential problem is then to find an
appropriate weight set for shortest path routing.
A simple way to set link weights is the hop-count method,
assigning the weight of each link to one. The length of a
path is thus the number of hops. Another default approach
recommended by Cisco is the inv-cap method (Cisco Systems
Inc., 2000;Thomas,2003), setting the weight of a link
inversely proportional to its capacity, without taking traffic
conditions into consideration. More generally, the weight of a
linkmaydependonandberelatedtoitstransmissioncapacity
and traffic load. Accordingly, a problem of interest is to find an
optimal weight set for shortest path routing (Burton and Toint,
1992), given a network topology, a projected traffic matrix
(Altın et al, 2010; Applegate and Cohen, 2006; Feldmann
et al, 2001;Wanget al, 2006), and an objective func-
tion (Balon et al , 2006;Pio
´
ro et al, 2002;Pio
´
ro and Medhi,
2004; Zhang, 2006).
The problem has two instances, depending on whether
multiple shortest paths or only a unique routing path from an
origin node to a destination node is allowed (Altin et al, 2013;
Bley et al, 2010; Giroire et al, 2015). For the first instance, a
number of heuristic methods have been introduced, each based
on, for example, a local search method mostly using an
increasing piecewise linear convex cost function or a heap-
reduction technique (Buriol et al, 2008; Fortz and Thorup,
2000, 2004
; Fortz and U
¨
mit, 2011; Ramalingam and Reps, 1996),
a genetic algorithm (Buriol et al, 2005; Ericsson et al, 2002;
Mulyana and Killat, 2002), simulated annealing (Pio
´
ro et al,
2002; Pio
´
ro and Medhi, 2004), Lagrangian relaxation (Holm-
berg and Yuan, 2000; Srivastava et al, 2005), an integrated
approach (Wang et al, 2001), or a MILP-based algo-
rithm (Amaldi et al, 2013; Cianfrani et al, 2012). For the second
instance, the Lagrangian relaxation method and local search
method have been proposed (Lin and Wang, 1993; Ramakrish-
nan and Manoel, 2001). These methods have been tested using
given data sets and have been verified to result in accept-
able routing performance. Meanwhile, with these heuristic
methods, the problem is not formulated exactly and is in general
This paper is based on a research project carried out at Imperial
College London. Some related results have been presented on
the 4th International Conference on Networking, Reunion
Island, April 2005, the 8th INFORMS Telecommunications
Conference, Dallas, USA, March 2006, INFORMS Annual
Meeting 2006, Pittsburgh, PA, USA, November 2006, and the
6th International Congress on Industrial and Applied
Mathematics, Zurich, Switzerland, July, 2007, respectively.
The author would like to thank the conference attendees of all the
corresponding sessions. An earlier version of this paper can be
found at http://arxiv.org/abs/0807.0038.
*Correspondence: Changyong Zhang, Department of Finance and
Banking, Faculty of Business, Curtin University Sarawak, Miri, Malaysia.
E-mail: changyong.zhang@curtin.edu.my
Journal of the Operational Research Society (2017) 68, 935–951
ª
2016 The Operational Research Society. All rights reserved. 0160-5682/17
www.palgrave.com/journals
not solved optimally. In particular, the resulting performance is
not consistently close to the optimal general routing (Bley et al,
2010; Fortz and Thorup, 2000). It is hence worth looking into the
possibility of formulating the problem explicitly, from which
optimal solutions may be obtained for data instances with
reasonable sizes arising from real-world applications.
From a management perspective, unique-path routing uses
simpler routing mechanisms and allows for easier monitoring
of traffic flows (Ben-Ameur and Gourdin, 2003; Hock et al,
2010). Hence, this paper considers the unique shortest path
routing problem, as specified in Section 2.1. It is a reduction in
the integer multi-commodity flow problem (Ahuja et al,
1993), which has been well addressed (Barnhart et al, 2000;
Dinitz et al, 1999; Park et al, 1996).
Partially due to the challenges involved in modeling the
problem appropriately, most existing solution algorithms are
heuristic (Bley, 2009; Kolliopoulos and Stein, 2001; Skutella,
2002). Efforts have been made to formulate the problem
mathematically. For example, a two-phase heuristic has been
proposed, to allocate a unique shortest path for each pair of
nodes and to compute link weights compatible with the set of
routing paths (Ben-Ameur and Gourdin, 2003). To guarantee
the existence of a compatible set of weights in the second
subproblem, necessary conditions are provided and discussed
in detail. The second problem is also referred to as the inverse
shortest paths problem, variants of which have been exten-
sively studied (Ahuja and Orlin, 2001, 2002; Bley, 2007;
Burton and Toint, 1992, 1994; Xu and Zhang, 1995; Zhang
and Liu, 1996), with or without the uniqueness of the
perceived optimal solution and the integrality of the perturbed
cost vector being taken into consideration.
Models without the necessity of resorting to the two-phase
heuristic have also been proposed (Zhang and Rodos
ˇ
ek,
2005a, b). This avoids considering the compatibility between
the two subproblems. In the meantime, as a critical step when a
model is introduced, the correctness of the models still remains
to be verified rigorously. Mathematical models have also been
developed for related problems (Bley and Koch, 2008; Farago
´
et al, 2003; Holmberg and Yuan, 2004), whereas they are
mostly path-based and potentially result in an exponential
number of constraints. This leaves space for further exploring
the structure properties of the problems, which may provide
more flexibility to derive alternative solution methods.
This paper focuses on mathematical modeling of the
problem, which may potentially yield a new exact solution
approach for real-world applications with average data sizes.
In particular, the correctness of the models is mathematically
proved rigorously. In Section 2, the problem is specified and
two different exact formulations are introduced. The second
one is then mathematically proved correct in Section 3.
Differences between the two formulations in both constraint
structure and model size are discussed in Section 4, followed
by the conclusion in Section 5.
The ideas behind the two formulations may be adopted to
model related problems in network routing and other fields. It
is also hoped that the steps of model formulation, model
validation, and model comparison may provide a reference
procedure for mathematical modeling.
2. Model formulation
2.1. Problem specifica tion
The unique shortest path routing problem is defined as follows.
Given
• a network topology, which is a directed graph structure
G¼ðN; LÞ, where
– N is a finite set of nodes, each of which represents a
router; and
– L is a set of directed links, each of which corresponds
to a transmission link; (For each ði; jÞ2L, i is the
starting node, j is the ending node, and c
ij
0 is the link
capacity.)
• a traffic matrix, which is a set of demands D; (It is assumed
that there is at most one demand between each origin–
destination pair. For each demand k 2D, s
k
2N is the
origin node, t
k
2N is the destination node, and d
k
[ 0is
the required bandwidth. Accordingly, S is the set of all
origin nodes, T
s
is the set of all destination nodes of
demands originating from node s 2S, and D
s
is the set of
all demands originating from node s 2S.)
• lower and upper bounds of link weights, which are positive
real numbers w
min
and w
max
, respectively; and
• an objective function, e.g., to maximize the sum of the
residual capacities,
find an optimal weight set w
ij
; ði; jÞ2L, subject to
• flow conservation constraints: For each demand, at each
node, the sum of all incoming flows (including the demand
bandwidth at the origin node) is equal to the sum of all
outgoing flows (including the demand bandwidth at the
destination node);
• link capacity constraints: For each link, the load of traffic
flows traversing the link does not exceed the capacity of
that link;
• path uniqueness constraints: Each demand has a unique
routing path; and
• path length constraints: For each demand, the length of
each path assigned to route the demand is strictly less than
that of any other possible and unassigned path to route the
demand.
By the above definition, the routing path of a demand is the
shortest one among all possible paths. For each link, the
routing path of a demand either traverses the link or not. The
path length and path uniqueness constraints require that the
length of the unique shortest path to route a demand is less
936
Journal of the Operational Research Society
Vol. 68, No. 8
than that of any other possible path from the origin to the
destination.
As shown in Figure 1, concerning the constraints, there are
three scenarios to be considered regarding the relationship
between the lengths of shortest paths and link weights.
• If the routing path of demand k traverses link (i, j), the
length of the shortest path from node s
k
to j is that from s
k
to i plus the weight of link (i, j);
• If the routing path of demand k does not traverse link
(i, j) but transits node j, the length of the shortest path from
node s
k
to j is strictly less than the sum of that from s
k
to
i and the weight of link (i, j); (otherwise, there would be at
least two shortest paths to route demand k.)
• If the routing path of demand k neither traverses link
(i, j) nor transits node j, the length of the shortest path from
node s
k
to j is less than or equal to the sum of that from s
k
to i and the weight of link (i, j).
With the problem being specified, below it is mathemati-
cally formulated from two different perspectives, based on the
study of the problem properties. For comparison, a demand-
based model is first introduced, followed by the origin-based
counterpart.
2.2. A demand-based model
Based on the observation on the relationship between the
length of a shortest path and the weights of links that it
traverses, the problem can be mathematically formulated as a
demand-based model (DBM) as follows, by defining one
routing decision variable for each link–demand pair (Zhang
and Rodos
ˇ
ek, 2005a).
• Routing decision variables:
x
k
ij
2f0; 1g; 8k 2D; 8ði; jÞ2L
ð1Þ
is equal to 1 if and only if the routing path of demand
k traverses link (i, j). The number of this set of variables is
jDjjLj.
• Link weight variables:
w
ij
2½w
min
; w
max
; 8ði; jÞ2L ð2Þ
denotes the routing cost of link (i, j). The number of this
set of variables is jLj.
• Path length variables:
l
s
i
2½0; þ1Þ; 8s 2S; 8i 2N ð3Þ
represents the length of the shortest path from origin node
s to node i. Apparently, l
s
k
t
k
is the length of the shortest path
to route demand k 2Dand l
s
s
¼ 0, 8s 2S. The number of
this set of variables is jSjjNj.
• Flow conservation constraints:
X
h:ðh;iÞ2L
x
k
hi
X
j:ði;jÞ2L
x
k
ij
¼
1; if i ¼ s
k
1; if i ¼ t
k
0; otherwise
8
>
<
>
:
;
8k 2D; 8i 2N:
ð4Þ
The number of this set of constraints is jDjjNj.
• Link capacity constraints:
X
k2D
d
k
x
k
ij
c
ij
; 8ði; jÞ2L:
ð5Þ
The number of this set of constraints is jLj.
• Path uniqueness constraints: under the combined restric-
tion of the flow conservation and path length constraints,
the constraints are satisfied automatically.
• Path length constraints:
x
k
ij
¼ 0 ^
X
h:ðh;jÞ2L
x
k
hj
¼ 0 ) l
s
k
j
l
s
k
i
þ w
ij
x
k
ij
¼ 0 ^
X
h:ðh;jÞ2L
x
k
hj
¼ 1 ) l
s
k
j
\l
s
k
i
þ w
ij
x
k
ij
¼ 1 ) l
s
k
j
¼ l
s
k
i
þ w
ij
9
>
>
>
>
>
=
>
>
>
>
>
;
;
8k 2D; 8ði; jÞ2L;
ð6Þ
which are in logic form, as interpreted in Section 2.1 and
illustrated in Figure 1 with thick lines being routing paths.
They can be linearized by introducing two constants e and
M with 0\e M. The new constraints are
l
s
k
j
l
s
k
i
þ w
ij
e
X
h:ðh;jÞ2L
x
k
hj
x
k
ij
0
@
1
A
l
s
k
j
l
s
k
i
þ w
ij
Mð1 x
k
ij
Þ
9
>
>
>
=
>
>
>
;
;
8k 2D; 8ði; jÞ2L:
ð7Þ
The number of this set of constraints is 2jDjjLj.By
enumerating all possible values of the routing decision
variables x
k
ij
; k 2D; ði; jÞ2L, it can be verified that the
linearized constraints are identical to the original ones. For
the efficiency in solving the problem, it is worth looking
i
j
h
s
k
t
k
w
ij
i
j
h
s
k
t
k
w
ij
ij
s
i
s
j
h
k
hj
k
ij
wllxx
kk
+≤=∧= 00
i
j
h
s
k
t
k
w
ij
ij
s
i
s
j
h
k
hj
k
ij
wllxx
kk
+<=∧= 10
ij
s
i
s
j
k
ij
wllx
kk
+== 1
Figure 1 Illustration of the path length constraints.
Changyong Zhang—An origin-based model for unique shortest path routing 937
into how to choose the values of e and M accordingly with
respect to the size of the network G¼ðN; LÞ as well as the
values of w
min
and w
max
.
• Objective function:
max
X
ði;jÞ2L
c
ij
X
k2D
d
k
x
k
ij
!
;
which is equivalent to
min
X
ði;jÞ2L
X
k2D
d
k
x
k
ij
:
ð8Þ
As a result, the complete model is
DBM :
Optimize ð8Þ
Subject to ð4Þ; ð5Þ; ð7Þ; ð1Þ; ð2Þ; ð3Þ
ð9Þ
A necessary condition of the unique shortest path routing
problem is the sub-path optimality requirement, which will be
invoked in the verification of the models in Section 3. The
requirement says that a sub-path of a routing path is also a
unique shortest path (Ben-Ameur and Gourdin, 2003; Bley
and Koch, 2008). Specifically, given an origin node s 2Sand
a node i 2N; i 6¼ s, all demands which originate from s and
transit i must go through the same incoming link to i.
Proposition 1 The path length constraints in the demand-
based model (9) imply the sub-path optimality constraints.
Proof Let k
1
; k
2
2D be two demands with s
k
1
¼ s
k
2
¼ s .
Assume that they use two disjoint paths to traverse from
node u to v. Demand k
1
uses path
P
i
¼ði
1
; i
2
Þ!ði
2
; i
3
Þ!!ði
m1
; i
m
Þ; ði
p
; i
pþ1
Þ
2L; p ¼ 1; ...; m 1;
where i
1
¼ u and i
m
¼ v, and demand k
2
uses path
P
j
¼ðj
1
; j
2
Þ!ðj
2
; j
3
Þ!!ðj
n1
; j
n
Þ; ðj
q
; j
qþ1
Þ
2L; q ¼ 1; ...; n 1;
where j
1
¼ u and j
n
¼ v.
By the definition of the routing decision variables,
x
k
1
i
p
i
pþ1
¼ 1 and x
k
2
i
p
i
pþ1
¼ 0; 8p 2f1; ...; m 1g:
Then by constraints (6), on the one hand, since
x
k
ij
¼ 1 ) l
s
k
j
¼ l
s
k
i
þ w
ij
, 8k 2D; 8ði; jÞ2L, considering
demand k
1
, it holds that
l
s
v
¼ l
s
u
þ l
u
i
m
¼ l
s
u
þ l
u
i
m1
þ w
i
m1
i
m
¼ l
s
u
þ l
u
i
m2
þ w
i
m2
i
m1
þ w
i
m1
i
m
¼ l
s
u
þ l
u
i
1
þ w
i
1
i
2
þþw
i
m1
i
m
¼ l
s
u
þ l
P
i
:
ð10Þ
On the other hand, since x
k
ij
¼ 0 ^
P
h:ðh;jÞ2L
x
k
hj
¼ 1 )
l
s
k
j
\l
s
k
i
þ w
ij
and x
k
ij
¼ 0 ) l
s
k
j
l
s
k
i
þ w
ij
, 8k 2D; 8ði; jÞ
2L, it follows from considering k
2
that
l
s
v
¼ l
s
u
þ l
u
i
m
\l
s
u
þ l
u
i
m1
þ w
i
m1
i
m
l
s
u
þ l
u
i
m2
þ w
i
m2
i
m1
þ w
i
m1
i
m
l
s
u
þ l
u
i
1
þ w
i
1
i
2
þþw
i
m1
i
m
¼ l
s
u
þ l
P
i
:
ð11Þ
Clearly, (10) and (11) contradict each other, which
means that the two demands cannot be routed over two
different paths between two shared nodes. It is hence
proved that the sub-path optimality constraints are satis-
fied. h
The sub-path optimality constraints are thus not explicitly
embedded into DBM. Mathematically, the constraints are
represented as
X
h:ðh;iÞ2L
max
k2D
s
x
k
hi
1; i 6¼ s; 8s 2S; 8i 2N;
ð12Þ
which can be linearized, with a new set of variables
y
s
ij
; s 2S; ði; jÞ2L,as
y
s
ij
x
k
ij
; 8k 2D
s
; 8ði; jÞ2L and
X
h:ðh;iÞ2L
y
s
hi
1; i 6¼ s; 8s 2S; 8i 2N:
By Proposition 1, the demand-based model defined in (9)is
equivalent to
Optimize ð8Þ
Subject to ð4Þ; ð5Þ; ð7Þ; ð12Þ; ð1Þ; ð2Þ; ð3Þ
ð13Þ
Model (13) will be invoked to verify the correctness of
DBM in Section 3.
2.3. An origin-based model
In Section 2.2, the unique shortest path routing problem is
formulated as a demand-based model, which defines one
routing decision variable for each link–demand pair. Based on
the study of properties associated with the solution, it can be
found that all routing paths of demands originating from the
same node constitute a tree, rooted at the origin node (Zhang
and Rodos
ˇ
ek, 2005b). Accordingly, a more natural formulation
is to define one routing decision variable for each link–origin
pair. For example, in Figure 2, instead of defining three
routing decision variables for link (i, j), one for each of the
three demands sharing the same origin node s, the new
938
Journal of the Operational Research Society
Vol. 68, No. 8
formulation defines only one routing decision variable for link
(i, j), paired with origin node s.
Based on the above observation, an origin-based model
(OBM) for the problem is formulated as follows.
• Routing decision variables:
y
s
ij
2f0; 1g; 8s 2S; 8ði; jÞ2L
ð14Þ
is equal to 1 if and only if the routing path of at least one
demand originating from node s traverses link (i, j). The
number of this set of variables is jSjjLj.
• Auxiliary flow variables:
f
s
ij
2½0; þ1Þ; 8s 2S; 8ði; jÞ2L
ð15Þ
corresponds to the load of traffic flows originating from
node s and traversing link (i, j). The number of this set of
variables is jSjjLj.
• Link weight variables:
w
ij
2½w
min
; w
max
; 8ði; jÞ2L ð16Þ
denotes the routing cost of link (i, j). The number of this
set of variables is jLj.
• Path length variables:
l
s
i
2½0; þ1Þ; 8s 2S; 8i 2N ð17Þ
represents the length of the shortest path from origin node
s to node i. In particular, l
s
s
¼ 0; 8s 2S. The number of this
set of variables is jSjjNj.
• Flow conservation constraints: For each tree, at the root
node, the difference between the sum of outgoing flows
and that of incoming flows is the sum of bandwidths of all
demands originating from the node; at the destination node
of each demand originating from the root node, the
difference between the sum of incoming flows and that of
outgoing flows is the bandwidth of the demand; and the
sum of incoming flows is equal to that of outgoing flows at
other nodes.
X
h:ðh;iÞ2L
f
s
hi
X
j:ði;jÞ2L
f
s
ij
¼
d
s
; if i ¼ s
d
k
; if i ¼ t
k
; 8k 2D
s
0; otherwise
8
>
<
>
:
; 8s 2S; 8i 2N;
ð18Þ
where d
s
¼
P
k2D
s
d
k
. The number of this set of constraints
is jSjjNj.
• Flow bound constraints: For each tree, the total flow load
over each link does not exceed the sum of all demand
bandwidths originating from the root node and it is equal to
zero if no demand originating from the root node is routed
through the link.
f
s
ij
y
s
ij
X
k2D
s
d
k
; 8s 2S; 8ði; jÞ2L:
ð19Þ
The number of this set of constraints is jSjjLj.
• Link capacity constraints:
X
s2S
f
s
ij
c
ij
; 8ði; jÞ2L:
ð20Þ
The number of this set of constraints is jLj.
• Path uniqueness constraints: For each tree, the number of
incoming links with nonzero flows is equal to zero at the
origin node, is equal to one at the destination node of a
demand originating from the root node, and does not
exceed one at any other node.
X
h:ðh;iÞ2L
y
s
hi
¼ 0; if i ¼ s
¼ 1; if i 2T
s
1; otherwise
8
>
<
>
:
; 8s 2S; 8i 2N: ð21Þ
The number of this set of constraints is jSjjNj.
• Path length constraints: For each tree, the length of the
unique shortest path to route a demand originating from the
root node is less than that of any other possible path from
the origin node to the destination node.
y
s
ij
¼ 0 ^
X
h:ðh;jÞ2L
y
s
hj
¼ 0 ) l
s
j
l
s
i
þ w
ij
y
s
ij
¼ 0 ^
X
h:ðh;jÞ2L
y
s
hj
¼ 1 ) l
s
j
\l
s
i
þ w
ij
y
s
ij
¼ 1 ) l
s
j
¼ l
s
i
þ w
ij
9
>
>
>
>
>
=
>
>
>
>
>
;
;
8s 2S; 8ði; jÞ2L:
The logic constraints can be linearized as
s
t
k3
t
k2
t
k1
j
i
Figure 2 Illustration of the origin-based model.
Changyong Zhang—An origin-based model for unique shortest path routing 939
