A comparison of routing sets for robust network design
Michael Poss
∗
Abstract
Designing a network able to route a set of non-simultaneous demand vectors is an important
problem arising in telecommunications. The problem can be seen a two-stage robust program
where the recourse function consists in choosing the routing for each demand vector. Allowing the
routing to change arbitrarily as the demand varies yields a very difficult optimization problem
so that different subsets of admissible routings have been discussed in the literature. In this
paper, we compare theoretically the optimal capacity allocation costs for six of these routing
sets: affine routing, volume routing and its two simplifications, the routing based on an arbitrary
2-cover of the uncertainty set, and the routing based on a cover delimited by a hyperplane. We
show that the two routing sets based on covers of the uncertainty set yield the same optimal
costs. We show then that the two simplified volume routings are special cases of affine routings.
Finally, assuming that the uncertainty set is the one studied by Bertsimas and Sim (2004), we
show that the optimal cost provided by volume routing is not less than the costs provided by
the simplified volume routings.
Keywords: Robust optimization; Network design; Routing set; Routing template; Affine rout-
ing.
1 Introduction
Given a graph and a set of point-to-point commodities with known demand values, the deterministic
network design problem aims at installing enough capacity on the arcs of the graph so that the
resulting network is able to route all commodities. In practice it is however very difficult to know
with precision the exact values of the demands at the time the design decisions are taken. In the
best case, we can estimate a set that contains most likely values for the demand. The introduction
of the uncertainty set leads to a robust optimization problem. In this context, a solution is said
to be feasible for the problem if it is feasible for all demand vectors that belong to the estimated
uncertainty set D, see Soyster [28] and Ben-Tal and Nemirovski [9, 10], among others. This rigid
framework is computationally easy but it does not allow the model to react against the uncertainty.
To address this drawback, Ben-Tal et al. [8] introduce two-stage robust optimization models that
allows to adjust a subset of the problem variables only after observing the actual realization of the
data. This adjusting procedure is often called recourse. This two-stage approach applies naturally
to network design since first stage capacity design decisions are usually made in the long term while
the routing decisions depend on the realization of the demand. Hence, the routing decisions can be
seen as the recourse. Free recourse is called dynamic routing in the context of robust network design
problems. It has been shown by Chekuri et al. [14] and Gupta et al. [18] that the robust network
design with dynamic routing is intractable. Already deciding whether or not a fixed capacity design
allows for a dynamic routing of demands in a given polytope is co-N P-complete (on directed graphs).
It is known already that two-stage robust programming with arbitrary recourse is computation-
ally intractable [8]. For this reason, Ben-Tal et al. [8] limit the recourse to affine functions of the
uncertainties which makes the problem tractable. Further works by Chen and Zhang [15] and Goh
and Sim [17] suggest to extend the second stage to piece-wise linear functions of the uncertainties. In
fact, considering special types of recourses had been used already in the context of network design.
Ben-Ameur and Kerivin [4, 5] introduce the concept of static routing: after fixing the design, the
routing of a commodity is allowed to change but only linearly with the variation of the commodity.
Static routing can also be seen as a single stage robust program where the set of routings paths to-
gether with the percental splitting among the paths are chosen at the same time the design decisions
∗
Department of Computer Science, Facult´e des Sciences, Universit´e Libre de Bruxelles, Brussels, Belgium,
mposs@ulb.ac.be
1
are made. The resulting set of paths and percental splitting is often called a routing template, which
is used by all demand vectors in the uncertainty set. The use of static routing makes the robust
network design problem tractable but it yields more expensive capacity allocations than the problem
with dynamic routing. Static routing has been used by various authors since its introduction by
Ben-Ameur and Kerivin, including Altin et al. [1], Koster et al. [19], Ord´o˜nez and Zhao [22].
Several authors tried to introduce routing schemes that are more flexible than static routing
while still being computationally easier than dynamic routing. Ben-Ameur [3] covers the demand
uncertainty set by two (or more) subsets using separating hyperplanes and uses specific routings
templates for each subset. The resulting optimization problem is N P-hard when no assumptions is
made on the hyperplanes. Scutell`a [27] generalizes this idea to arbitrary covers of the uncertainty
set. She allows a set of routing templates to be used conjointly so that each demand vector can be
routed by at least one of the routing templates. She also introduces a procedure that works in two
steps. First, an optimal capacity allocation with static routing is computed. Then, she allows allows
to reroute part of the demand vectors according to a second routing template. Ben-Ameur and
Zotkiewicz [6] introduce volume routing, a framework that shares the demand between two routing
templates, according to thresholds. They prove that the resulting optimization problem is N P-
hard and introduce two simplifications. Finally, applying the affine recourse from Ben-Tal et al. [8]
to robust network design problems, Ouorou and Vial [24] introduce the concept of affine routing.
Recently, Poss and Raack [26, 25] study the properties of affine routing, and compare the later to the
static and dynamic routings, both theoretically and empirically. They conclude that affine routing
tends to yield very good approximations of dynamic routing while being computationally tractable.
In this paper, we compare theoretically the optimal capacity allocation costs provided by the affine
routings from Ouorou and Vial [24], the volume routings from Ben-Ameur and Zotkiewicz [6], and
the routings based on covers of the uncertainty set in two subsets (Ben-Ameur [3] and Scutell`a [27]).
In the next section, we introduce the robust network design problem and define a routing set. We
model the robust network design problem with the explicit dependency on the routing set and
formalize each of the routing frameworks studied herein. Our main results are stated in Section 2.3.
In Section 3, we try to understand how good is the cost of the optimal capacity allocation provided
by each of the routing sets, and we compare these costs among the different routing sets. We start
by comparing the costs obtained with routings based on covers of the uncertainty set in Section 3.2.
Then, in Section 3.3, we turn to affine and volume routings. In Section 4, we present examples
showing that it is not possible, in general, to compare some of these costs. Finally, we conclude the
paper in Section 5.
2 Robust network design
2.1 Problem formulation
The problem is defined below for a directed graph G = (V, A) and a set of commodities K. We
formalize first the concept of a routing. Then, we introduce the robust network design problem.
Each commodity k ∈ K has a source s(k) ∈ V , a destination t(k) ∈ V , and a demand value d
k
≥ 0.
A multi-commodity flow is a vector f ∈ R
|A|×|K|
+
that satisfies the flow conservation constraints at
each node of the network:
X
a∈δ
+
(v)
f
k
a
−
X
a∈δ
−
(v)
f
k
a
=
d
k
if v = s(k)
−d
k
if v = t(k)
0 else
for each v ∈ V, (1)
where δ
+
(v) and δ
−
(v) respectively denote the set of outgoing arcs and incoming arcs at node v.
In this work the values of the demand vector are uncertain and belong to the closed, convex, and
bounded set D ⊂ R
|K|
+
. We call such a set an uncertainty set and any d ∈ D is called a realization
of the demand. We denote by (D, R
|A|×|K|
) the set of all functions from D to R
|A|×|K|
. Then, a
routing is a function f ∈ (D, R
|A|×|K|
) that satisfies (1) for all realizations of the demand, that is
X
a∈δ
+
(v)
f
k
a
(d) −
X
a∈δ
−
(v)
f
k
a
(d) =
d
k
if v = s(k)
−d
k
if v = t(k)
0 else
for all v ∈ V, d ∈ D, (2)
and that is non-negative
f
k
a
(d) ≥ 0 for all d ∈ D. (3)
2
A routing with no further restrictions is called dynamic routing. Hence, the set of all dynamic
routings is the set of all functions from D to R
|A|×|K|
that satisfy (2) and (3):
F ≡
n
f ∈ (D, R
|A|×|K|
) | f satisfies (2) and (3)
o
. (4)
In this paper, we are interested in using special kinds of routings. This corresponds to using specific
subsets F
0
⊆ F. These subsets are described in the next section. In what follows, we describe the
robust network design problem making explicit the set of admissible routings.
A vector x ∈ R
|A|
+
is called a capacity allocation. A capacity allocation is said to support the set
D if there exists a dynamic routing f ∈ F serving D such that for every d ∈ D the corresponding
multi-commodity flow f(d) does not exceed the capacities described by x. Similarly, we say that
(x, f ) supports D when both the routing f and the capacity allocation x are given. More generally,
we say that (x, F
0
) supports D when there exists a routing f ∈ F
0
such that (x, f ) supports D.
Given an uncertainty set D and a routing set F
0
⊆ F, robust network design now aims at providing
the cost minimal capacity allocation x such that (x, F
0
) supports D:
min
X
a∈A
κ
a
x
a
RND(F
0
) s.t. f ∈ F
0
(5)
X
k∈K
f
k
(d) ≤ x, d ∈ D (6)
x ≥ 0,
where κ
a
∈ R is the cost for installing one unit of capacity on arc a ∈ A. Notice that in real
applications, these costs are usually non-negative. We shall denote the optimal cost of RND(F
0
)
by opt(F
0
). Problem RND(F
0
) contains an infinite number of variables f (d) for all d ∈ D as well
as an infinite number of capacity constraints (6). Moreover, the problem may not even be linear,
depending on the constraints defining set F
0
.
Considering the set of all routings F, RND(F) is a two-stage robust program with recourse
following the more general framework described by Ben-Tal et al. [8]. The capacity design has to
be fixed in the first stage, and observing a demand realization d ∈ D, we are allowed to adjust the
routing f (d) arbitrarily in the second stage. In that case, (5) is replaced by (2) and (3) so that
RND(F) is a linear program, yet infinite. Whenever D is a polytope, Poss and Raack [26], among
others, show how to provide a finite linear programming formulation for RND(F). The formulation
is based on enumerating the extreme points of D, so that its size tends to increase exponentially with
the number of commodities. In fact, the problem is very difficult to solve given that only deciding
whether a given capacity allocation vector x supports D is coN P-complete for general polytopes D,
see Chekuri et al. [14] and Gupta et al. [18]. Moreover, the use of dynamic routings suffers from
another drawback. It may be difficult in practice to change arbitrarily the routing according to the
demand realization.
For these reasons, various authors study restrictions on the routings that can be used, introducing
different subsets of routings F
0
⊂ F. Their hope is that opt(F
0
) provides a good approximation of
opt(F) while yielding an easier optimization problem RND(F
0
). For instance, Frangioni et al. [16]
and Poss and Raack [26] show under very strong assumptions on D that the optimal capacity al-
locations provided by dynamic routings are equivalent to the ones provided by static routings and
affine routings, respectively, which are polynomially solvable when D has a compact formulation. In
the next section, we present different choices of F
0
discussed in the literature, including static and
affine routings. Then, we summarize the main contributions of this paper in Section 2.3.
Note that if there exists only one path from s(k) to t(k) for a commodity k ∈ K, then all routings
coincide for that commodity. Unless stated otherwise, in the following we assume that for all k ∈ K
there exist at least two distinct paths p
1
, p
2
in G from s(k) to t(k), that is, two paths that differ by
one arc at least.
2.2 Routings frameworks
In the next sections, we define formally the set of static routings and the routing sets from Ouorou
and Vial [24], Ben-Ameur [3], Scutell`a [27] and Ben-Ameur and Zotkiewicz [6].
3
2.2.1 Static routing
The simplest alternative to dynamic routing has been introduced by Ben-ameur [5] and has been
used extensively since then, see Altin et al. [1, 2], Koster et al. [19], Mudchanatongsuk et al. [21],
and Ord´o˜nez and Zhao [22]. This framework considers a restriction on the second stage recourse
known as static routing (also called oblivious routing). Each component f
k
: D → R
|A|
+
is forced to
be a linear function of d
k
:
f
k
a
(d) := y
k
a
d
k
a ∈ A, k ∈ K, d ∈ D. (7)
Notice that (7) implies that the flow for k is not changing if we perturb the demand for h 6= k. By
combining (2) and (7) it follows that the multipliers y ∈ R
|A|×|K|
+
satisfy to
X
a∈δ
+
(v)
y
k
a
−
X
a∈δ
−
(v)
y
k
a
=
1 if v = s(k)
−1 if v = t(k)
0 else
for each v ∈ V. (8)
The flow y is called a routing template since it decides, for every commodity, which paths are used
to route the demand and what is the percental splitting among these paths. We define formally the
the set of all routing templates as
Y ≡
n
y ∈ R
|A|×|K|
+
| y satisfies (8)
o
, (9)
and the set of all static routings as
F
stat
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y ∈ Y : f
k
a
(d) = y
k
a
d
k
a ∈ A, k ∈ K, d ∈ D
o
.
An important result is that a compact linear formulation can be provided for RND(F
stat
) as long as
the description of D is compact (see Altin et al. [2] among others). Hence, the resulting optimization
problem is polynomially solvable.
In the following, we review alternative routing sets F
0
that are less restrictive than static routings
while not being as flexible as dynamic routings. Said differently, F
stat
⊆ F
0
⊆ F.
2.2.2 Covers of the uncertainty set delimited by a hyperplane
Given a set D, a collection of subsets of D forms a cover of D if D is a subset of the union of sets in
the collection. Ben-Ameur [3] introduces the idea of covering the uncertainty set by two (or more)
subsets using hyperplanes and proposes to use a routing template for each subset. This yields the
following set of routings:
F
2|
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y
1
, y
2
∈ Y and α ∈ R
K
, β ∈ R :
f
k
a
(d) =
y
1k
a
d
k
d ∈ D ∩ {d, αd ≤ β}
y
2k
a
d
k
d ∈ D ∩ {d, αd ≥ β}
a ∈ A, k ∈ K, d ∈ D
.
The definition above implies that both routing templates y
1
and y
2
must be able to route demand
vectors that lie in the hyperplane {d, αd = β} without exceeding the capacity. He proves that
RND(F
2|
) is N P-hard in general and describes simplification schemes, where α is given. He further
works on the framework in Ben-Ameur and Zotkiewicz [7].
2.2.3 Arbitrary covers of the uncertainty set
Scutell`a [27] introduces the idea of using conjointly two routing templates. Formally, she proposes
to use two routing templates y
1
and y
2
such that each d ∈ D can be served either by y
1
or by y
2
(or
both). This yields the following set of routings:
F
2
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y
1
, y
2
∈ Y and D
1
, D
2
⊆ D, D = D
1
∪ D
2
:
f
k
a
(d) =
y
1k
a
d
k
d ∈ D
1
y
2k
a
d
k
d ∈ D
2
a ∈ A, k ∈ K, d ∈ D
.
4
She mentions that the complexity of RND(F
2
) is unknown. We show in this paper that this
optimization problem is N P-hard, because it is a generalization of RN D(F
2|
), proved to be N P-
hard by Ben-Ameur [3]. The framework described by F
2
has been independently proposed for general
robust programs by Bertsimas and Caramanis [11] (see also Bertsimas et al. [12]) where the authors
propose to cover the uncertainty sets with k subsets and devise independent sets of recourse variables
for each of these subsets.
2.2.4 Volume routings
More recently, Ben-Ameur and Zotkiewicz [6] introduce a framework that shares the demand between
two routing templates, according to thresholds h
k
for each k ∈ K. Formally, they use the following
set of routings:
F
V
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y
1
, y
2
∈ Y, h ∈ R
K
+
:
f
k
a
(d) = y
1k
a
min(d
k
, h
k
) + y
2k
a
max(d
k
− h
k
, 0) a ∈ A, k ∈ K, d ∈ D
.
They prove that RND(F
V
) is an N P-hard optimization problem. Hence, they introduce simpler
frameworks described below. Defining d
k
min
= min
d∈D
d
k
and d
k
max
= max
d∈D
d
k
, the set of routings
becomes one of the following
F
VS
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y
1
, y
2
∈ Y : f
k
a
(d) = y
1k
a
d
k
min
+ y
2k
a
(d
k
− d
k
min
) a ∈ A, k ∈ K, d ∈ D
o
,
F
VG
≡
n
f ∈ (D, R
|A|×|K|
) | ∃y
1
, y
2
∈ Y :
f
k
a
(d) = y
1k
a
d
k
min
d
k
max
− d
k
d
k
max
− d
k
min
+ y
2k
a
d
k
max
d
k
− d
k
min
d
k
max
− d
k
min
a ∈ A, k ∈ K, d ∈ D
,
which are both well-defined whenever d
k
min
< d
k
max
for each k ∈ K. When d
k
min
= d
k
max
for some
k ∈ K, the k-th component of f ∈ F
VG
is defined by f
k
(d) = y
1k
d
k
.
2.2.5 Affine routings
Ben-Tal et al. [8] introduce Affine Adjustable Robust Counterparts restricting the recourse to be an
affine function of the uncertainties. Ourou and Vial [24] apply this framework to robust network
design by restricting f
k
to be an affine function of all components of d giving
F
aff
≡ {f ∈ (D, R
|A|×|K|
)|∃f
0
∈ R
K
, y ∈ R
|A|×|K|
:
f
k
a
(d) = f
0k
a
+
X
h∈K
y
kh
a
d
h
a ∈ A, k ∈ K, d ∈ D, f satisfies (2) and (3)
)
.
This framework has been compared theoretically and numerically to static and dynamic routings by
Poss and Raack [26]. In particular, the authors show that a compact formulation can be described
for RN D(F
aff
) as long as D has a compact description, generalizing the result obtained for static
routing already. We point out that a major difference between F
aff
and the routing described in
Section 2.2.1-2.2.4 is that the formers are build up using routing templates, so that it is implicitly
assumed that flow conservation constraints (2) and non-negativity constraints (3) are satisfied. In
opposition, routings in F
aff
are build up using ordinary vectors so that that satisfaction of (2) and
(3) must be stated explicitly.
2.3 Contributions of this paper
The objective of this paper is to compare opt(F
0
) among the routing sets recalled in previous sections.
This comparison is carried out in Section 3. Our main results are stated next.
(a) Let D be an uncertainty set. It holds that opt(F
2
) = opt(F
2|
), opt(F
aff
) ≤ opt(F
VG
) ≤
opt(F
VS
), and opt(F
V
) ≤ opt(F
VS
) for any cost vector κ ∈ R
|A|
.
(b) Let D be an uncertainty polytope such that for each k ∈ K, there exists non-negative numbers
0 ≤ d
k
min
≤ d
k
max
such that d
k
∈ {d
k
min
, d
k
max
} for each extreme point of D. It holds that
opt(F
V
) = opt(F
VS
) for any cost vector κ ∈ R
|A|
.
5