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A comparison of dierent routing schemes for the robust
network loading problem: polyhedral results and
computation
Sara Mattia, Michael Poss
To cite this version:
Sara Mattia, Michael Poss. A comparison of dierent routing schemes for the robust network load-
ing problem: polyhedral results and computation. Computational Optimization and Applications,
Springer Verlag, 2018, 69 (3), pp.753 - 800. �10.1007/s10589-017-9956-z�. �hal-01768646�
A comparison of different routing schemes for the robust
network loading problem: polyhedral results and computation
Sara Mattia
∗
Michael Poss
†
Abstract
We consider the capacity formulation of the Robust Network Loading Problem. The aim of the
paper is to study what happens from the theoretical and from the computational point of view
when the routing policy (or scheme) changes. The theoretical results consider static, volume,
affine and dynamic routing, along with splittable and unsplittable flows. Our polyhedral study
provides evidence that some well-known valid inequalities (the robust cutset inequalities) are
facets for all the considered routing/flows policies under the same assumptions. We also intro-
duce a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead,
they are facets in some settings, but not in others. A branch-and-cut algorithm is also proposed
and tested. The computational experiments refer to the problem with splittable flows and the
budgeted uncertainty set. We report results on several instances coming from real-life networks,
also including historical traffic data, as well as on randomly generated instances. Our results
show that the problem with static and volume routing can be solved quite efficiently in practice
and that, in many cases, volume routing is cheaper than static routing, thus possibly repre-
senting the best compromise between cost and computing time. Moreover, unlikely from what
one may expect, the problem with dynamic routing is easier to solve than the one with affine
routing, which is hardly tractable, even using decomposition methods.
Keywords: robust network loading, budgeted uncertainty, Benders decomposition, static rout-
ing, volume routing, affine routing, dynamic routing.
1 Introduction
Given an undirected graph and a set of point-to-point commodities with known demands (traffic
matrix), the objective of the network design problem is to find the cheapest capacity installation on
the edges of the graph such that the resulting network supports the routing of the commodities. The
problem has numerous applications in telecommunications, transportation, and energy management,
among many others. Accordingly, a large number of variations can be defined, which restrict for
instance, the type of flows admissible on the edges or the type of capacities that can be installed on
the edges. A large variety of technical constraints can also be considered. They include ensuring a
given level of survivability in case of link failures or limiting the length of the paths used. Different
flows policies may be used: the demands may be restricted to be routed on single paths (unsplit-
table flows) or the flows may be unrestricted (splittable flows) [9]. Herein, we focus on the so-called
Network Loading Problem, where capacities can be installed by integer multiples. An important
aspect of network design problems is related to the knowledge of the demands. In a large number
of applications, these demands are not available at the time we decide on the capacity installation.
Here we refer to the Network Loading Problem under uncertain demands as the Robust Network
Loading Problem. A way to deal with demand uncertainty is to rely on demand forecasts, based, for
instance, on population statistics [18] or on traffic measurements [48]. If these statistical studies are
accurate enough, one can come up with a stochastic model that considers demands as known random
∗
S. Mattia. Istituto di Analisi dei Sistemi ed Informatica. Consiglio Nazionale delle Ricerche. Via dei Taurini 19,
00185 Roma (Italy). Email: sara.mattia@iasi.cnr.it.
†
M. Poss. UMR CNRS 5506 LIRMM. Universit´e de Montpellier. Rue Ada 161, 34095 Montpellier cedex 5, (France).
Email: michael.poss@lirmm.fr. Corresponding author.
1
variables, typically leading to two-stage stochastic programs. See [5] and the references therein for
additional details. Unfortunately, it is very difficult in practice to come up with an accurate de-
scription of these random variables. An alternative approach is to define sets of admissible values of
random variables compatible with the available data, falling into the framework of distributionally
robust optimization [46]. In this work, we assume that the uncertain demands are described through
an uncertainty set [11], falling into the framework of robust optimization [13]. Hence, the problem
turns to designing a network able to route each traffic matrix in the uncertainty set. Although
conservative, this approach has been used extensively in recent years to model demand uncertainty
in telecommunications and transportation networks [4, 8, 26, 33, 34, 37, 38, 44]. A popular choice to
model the forecast demands is to use the budgeted uncertainty set [15]. The latter supposes that the
demands fluctuate between their nominal values and given peak values and that at most Γ of them
reach their peak values simultaneously. The model is motivated by probabilistic guarantees [15] and
has been used in numerous papers on robust network design problems [8, 34, 44].
The introduction of uncertainty in demands raises the question of how to adapt the flows to dif-
ferent realizations of the demand. This concept is often referred to as routing in the literature on
network optimization. Different routing policies (or schemes) have been studied in the past, each
with its own flexibility and computational issues [4, 33, 44]. At one extreme, we find static routing,
which imposes that the fractional splitting of the commodities among a fixed set of paths stays
constant for all realizations of the demands. In the other extreme, dynamic routing allows the flows
to be changed completely any time that the demands change. Both approaches have advantages and
drawbacks. Dynamic routing is more flexible, but the corresponding problem is difficult to solve.
In addition, dynamic routing can be difficult to implement in practice because the routing depends
on the current status of all the demands in the network, thus, hardening decentralization. On the
other hand, static problems are usually computationally more tractable and easier to implement in
decentralized environments, but the corresponding solutions may be too conservative. Therefore, in
the last years, several intermediate routing schemes have been proposed. They include affine and
volume routing. The aim was to obtain more flexibility than static routing, solving a problem that is
theoretically easier than the dynamic one. Affine routing [41] restricts the flows to be affine functions
of the demands, as it applies to network optimization what has long been known as affine decision
rules in adjustable robust optimization [14]. Affine routing has been used in several papers on robust
network design, see for instance [8, 28, 40, 44], including a variation of the problem where it is the
capacity, rather than the demand, which is uncertain [42]. Volume routing [12] is a special case of
affine routing, where the set of paths used for each commodity can be adjusted according to the
current value of the demand for that commodity. In addition to its numerical tractability, volume
routing is easier to implement in a decentralized environment than affine and dynamic routing, since
the routing for a commodity only depends on the demand for that commodity. Other intermediate
routings have also been proposed in the literature, such as those based on dynamic partitions of the
uncertainty set [10, 45]. However, they lead to optimization problems that are even harder to solve
than the problem with dynamic routing [43]. For this reason, we do not consider them in what follows.
The computational tractability of the Robust Network Loading Problem has been studied under
static, affine and dynamic routing. The problem is N P-hard independently of the routing scheme,
as it includes the problem without uncertainty, and then the Steiner tree problem, as a special case.
However, even for the splittable case, when the integrality restrictions on the capacity variables are
relaxed, the complexity of the problem does depend on the routing. For affine, volume and static
routing there exists a compact formulation, that is, a formulation with a polynomial number of
variables and constraints, and, hence, the problem can be solved in polynomial time [3, 12, 44]. In
contrast, the problem with dynamic routing was proved to be N P-hard, both in general and for the
budgeted uncertainty set [20, 25, 36]. Therefore, no compact formulation exists, unless P = N P.
As a consequence, for static/affine/volume routing separation is polynomial, whereas for dynamic
routing it is N P-hard [33]. This affects the computational performance of the problem with integer
capacities. For splittable flows and static routing, in [4] the authors propose a polyhedral investiga-
tion and numerical results for the problem assuming that the demand uncertainty polytope is the
Hose model [22, 23]. Similar numerical studies have been carried out in [26] for the problem with
2
budgeted uncertainty. In [27] a Benders decomposition approach for the splittable Robust Network
Loading Problem with static routing and budgeted uncertainty is derived and studied. In [33] the
author studies the problem with dynamic routing and splittable flows under the Hose model, propos-
ing a branch-and-cut procedure related to bilevel optimization. A special case of that problem is
studied in [19], where the authors focus on single-commodity problems, which allows them to provide
more efficient inequalities and separation routines. Finally, in [37] the problem with splittable flows,
affine routing and polyhedral or ellipsoidal uncertainty sets is studied. The authors further consider
a version of the problem with restrictions on the set of feasible paths and propose column generation
algorithms.
In this paper we study the capacity formulation of the Robust Network Loading Problem, that is, a
formulation including only design variables. The scope of the paper is to investigate what happens
both theoretically and computationally when the routing policy changes. To the best of our knowl-
edge, this is the first time that such a comprehensive investigation is carried out. From the theoretical
point of view, a first contribution of the paper is to derive a capacity formulation for the problem
with volume and affine routing using a Benders decomposition approach. For dynamic routing, the
exponential number of variables and constraints in the formulation prevents us from using the flow
formulation as a starting point for deriving a capacity formulation, and hence, from applying the
classical Benders reformulation. Instead, we must draw from the more advanced tools proposed in
the recent years for adjustable robust optimization, see [7, 33, 47]. We refer to the formulations in-
cluding only design variables as Benders formulations in the rest of the paper, despite the differences
in the techniques used to obtain them. We provide polyhedral results characterizing the convex-hull
of integer feasible solutions of the problem for all the considered routing schemes under splittable
and unsplittable flows, for a general uncertainty set. We establish relations among the polyhedra
corresponding to the considered routing schemes, giving conditions for an inequality that is valid
or facet under a given routing/flows policy pair to be valid or facet for another routing/flows pair.
Following the comments in [33], we formally prove that the well-known robust cutset inequalities
[26, 33, 34] are facets under the same assumptions in all the considered settings. Actually, this is,
so far, the only class of inequalities having such a property, but for the non-negativity constraints.
Indeed, we present a new class of valid inequalities for the Robust Network Loading Problem, the
robust 3-partition inequalities, and show that they are facet-defining for the problem with dynamic
routing and splittable flows, whereas they are not facets, under the same assumptions, for the other
routing schemes or for unsplittable flows. They provide the first example of such a behavior. From
a computational perspective, we investigate what is the effect of the different routing schemes on
costs and computing times, when the splittable problem is solved using the budgeted uncertainty
set, the Benders formulations and the cuts investigated in the theoretical part. We provide exact
and heuristic separation routines for the robust cutset inequalities, a heuristic approach for finding a
violated robust 3-partition inequality, an exact approach for the Benders cuts and a primal heuristic.
We compare the Benders formulations using many different cutting plane approaches. We also in-
vestigate the compact flow formulation for volume, affine and static routing. Although compact, the
corresponding problem may be time consuming for some of the considered routing schemes. How-
ever, we show that its performance can be significantly improved by separating the above mentioned
inequalities in the branch-and-cut tree. We report computational experiments on real-life instances,
including instances based on historical traffic data [39], as well as on randomly generated ones. We
prove that the problem with static and volume routing can be solved quite efficiently in practice on
the considered instances. Moreover, unlikely from what one may expect, the problem with dynamic
routing is easier to solve than the one with affine routing. It turns out that all the considered routing
schemes (but possibly affine routing) are able to solve real-life problems. We also show that volume
routing yields cost reductions over static routing in half of the instances, while not requiring more
computational time. This is important because many papers use static routing, whereas volume
routing is not so popular.
The paper is structured as follows. Sections 2 and Section 3 present, respectively, the so-called flow
and Benders formulations of the Robust Network Loading Problem for each of the aforementioned
routing schemes. In Section 4 we give polyhedral results. In Section 5 we present implementation
3
details and the test-bed. In Sections 6 we discuss computational experiments. The paper is concluded
in Section 7. When presenting models and computational results (Sections 2, 3, 5, 6), we restrict to
splittable flows and to the budgeted uncertainty set, whereas the theoretical results (Section 4) are
more general and they also hold for unsplittable flows and are independent of the uncertainty set.
2 Flow formulations
Let G(V, E) be an undirected graph without loops and parallel edges, let K be the set of point-to-
point commodities to be routed on the network and assume that all the demands belonging to a
given uncertainty set U ⊂ R
|K|
+
must be served. Each commodity k ∈ K is defined by its endnodes
s
k
and t
k
and its demand value d
k
for any d ∈ U. In presenting the models, we suppose that the
flows are splittable and that the uncertainty set has a special structure, often used in the literature
[15]: each demand value d
k
varies between its nominal value
¯
d
k
and its peak value
¯
d
k
+
ˆ
d
k
and the
number of deviations from the nominal value is bounded by integer Γ. It corresponds to the extended
formulation below and we denote it by U
b
.
U
b
≡
(
d ∈ R
|K|
+
| ∃δ ∈ [0, 1]
|K|
: d
k
=
¯
d
k
+ δ
k
ˆ
d
k
, k ∈ K,
X
k∈K
δ
k
≤ Γ
)
(1)
In what follows we use U when the formulation or the result refers to a general uncertainty set,
whereas we use U
b
when we specifically refer to the budgeted uncertainty set. We do not make any
special assumption on U, but that it is bounded (otherwise the resulting problem is unbounded)
and non-empty (otherwise the resulting problem is trivial). By non-empty we also mean that U is
different from the singleton {0}, as the resulting problem admits the trivial solution x = 0, as well.
We associate to E the set of directed arcs A: for each e = {i, j} ∈ E, we create two directed arcs
(i, j) and (j, i). The unitary cost of installing capacity on edge e ∈ E is given by c
e
. The Robust
Network Loading Problem studied herein aims at installing the cheapest capacities x on the edges
of the graph, such that all realizations of the demand vectors d ∈ U can be routed on the resulting
network.
2.1 Dynamic routing
The problem with dynamic routing can be formulated mathematically as follows. The integer variable
x
e
represents the capacity allocation on edge e ∈ E and the real variable f
k
ij
(d) describes the amount
of flow for commodity k routed on arc (i, j) ∈ A, when considering demand d ∈ U. We also define
the star of node i ∈ N as N (i) = {j ∈ N : ∃e = {i, j} ∈ E}.
fDRNL min
X
e∈E
c
e
x
e
s.t.
X
j∈N(i)
(f
k
ji
(d) − f
k
ij
(d)) =
d
k
if i = t
k
0 otherwise
i ∈ V \ {s
k
},
k ∈ K, d ∈ U
(2a)
X
k∈K
(f
k
ij
(d) + f
k
ji
(d)) ≤ x
e
e = {i, j} ∈ E, d ∈ U (2b)
f, x ≥ 0 (2c)
x ∈ Z
|E|
(2d)
Constraints (2a) represent flow conservation constraints at every node of the network (constraints
for s
k
are not included because they are redundant) and constraints (2b) impose that the amount
of flow on each edge does not exceed the available capacity on that edge. In the following, we call
routing the flow function f. We use the term dynamic routing when no particular assumption is made
on admissible functions, as in f DRNL. Problem fDRNL is a mixed integer linear programming
problem with an infinite number of variables and constraints. However, we see easily that the problem
can be discretized by considering the extreme points of the demand polyhedron, denoted by vert(U),
4
