2011/37
■
Benders decomposition for the hop-constrained
survivable network design problem
Quentin Botton, Bernard Fortz,
Luis Gouveia and Michael Poss
Center for Operations Research
and Econometrics
Voie du Roman Pays, 34
B-1348 Louvain-la-Neuve
Belgium
http://www.uclouvain.be/core
DISCUSSION PAPER
CORE DISCUSSION PAPER
2011/37
Benders decomposition for the hop-constrained
survivable network design problem
Quentin BOTTON
1
, Bernard FORTZ
2
,
Luis GOUVEIA
3
and Michael POSS
4
July 2011
Abstract
Given a graph with nonnegative edge weights and node pairs Q, we study the problem of constructing a
minimum weight set of edges so that the induced subgraph contains at least K edge-disjoint paths
containing at most L edges between each pair in Q. Using the layered representation introduced by Gouveia
(1998), we present a formulation for the problem valid for any K, L ≥ 1. We use a Benders decomposition
method to efficiently handle the big number of variables and constraints. We show that our Benders cuts
contain the constraints used by Huygens et al. to formulate the problem for L = 2,3,4, as well as new
inequalities when L ≥ 5. While some recent works on Benders decomposition study the impact of the
normalization constraint in the dual subproblem, we focus here on when to generate the Benders cuts. We
present a thorough computational study of various branch-and-cut algorithms on a large set of instances
including the real based instances from SNDlib. Our best branch-and-cut algorithm combined with an
efficient heuristic is able to solve the instances significantly faster than CPLEX 12 on the extended
formulation.
Keywords: network design, survivability, hop-constraints, benders decomposition, branch-and-cut
algorithms.
Mathematics Subject Classification (2000): 90C27, 68M10, 90C57, 49M27, 90C10.
1
Université catholique de Louvain, CESCM-LSM and CORE, B-1348 Louvain-la-Neuve, Belgium.
E-mail: Quentin.botton@uclouvain.be
2
GOM, Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles, Belgium.
E-mail: bfortz@ulb.ac.be
3
Faculdade de Ciências da Universidade de Lisboa, DEIO, P-1749-016 Lisbon, Portugal. E-mail: legouveia@fc.ul.pt
4
GOM, Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles, Belgium.
E-mail: mposs@ulb.ac.be
The authors would like to thank the Associate Editor and the three referees for their comments and suggestions that help
in improving the presentation of this paper. This research is supported by an “Actions de Recherche Concertées” (ARC)
projet of the “Communauté française de Belgique” and a FRFC program “Groupe de Programmation Mathémathique”.
Michael Poss is a research fellow of the “Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture”
(FRIA). Part of this research was conducted while Quentin Botton visited the University of Lisbon, supported by the
“Concours Bourses de Voyage” of the “Communauté française de Belgique” and Marie-Curie Research Training
Network “ADONET”. The work of Luis Gouveia is supported by the grant MATH-LVT-Lisboa-152.
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian
State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.
1. Intro duction
1.1 Problem motivation
Our worldwide society is largely dependent on the performance of huge information systems which
are often organized in large-scale, complex and costly networks. With time, the equipment (routers,
fiber optic cables, ...) deteriorates and the risk of failure must be controlled as well as possible by
the network managers in order to guarantee the best service to users. As a consequence, the
development of survivable networks became a crucial field of research and investigation. In this
paper, we define a survivable network as a network in which the various demands can be routed
without loss of service quality even in case of network failure (link or node failure).
For each demand, we impose that at least K different paths exist for each origin-destination
pair. These K paths can, for instance, be “edge-disjoint”, i.e. if a particular edge belongs to one
path, this particular edge cannot be used by the other K − 1 paths. This guarantees that if K − 1
edges break down, it is always possible to reroute all the demands by the K-th path which does not
use the broken arcs. Another form of survivability considers the node-disjoint case. More formally
consider an undirected graph G = (V, E), where V represents the vertex set, and E the set of edges.
We also associate an installation cost c
ij
to each edge ij and introduce an auxiliary arc set A which is
obtained from each edge ij of E by creating two arcs (i, j) and (j, i) with the same cost as the original
edge. In order to incorporate the survivability considerations into the problem definition, we need
to introduce the following graph theoretical concepts with elements from the sets V and A. Given
two distinct nodes o (the origin vertex of demand) and d (the destination vertex of demand) of V ,
an od-path is a sequence of node-arcs P = (v
0
, (v
0
, v
1
), v
1
, ..., (v
l−1
, v
l
), v
l
), where l ≥ 1, v
0
, v
1
, ..., v
l
are distinct vertices, v
0
= o, v
l
= d, and (v
i−1
, v
i
) is an arc connecting v
i−1
and v
i
(for i = 1, ..., l).
A collection P
1
, P
2
, ..., P
k
of od-paths is called edge-disjoint if any arc (i, j) and its symmetric arc
(j, i) appears in at most one path. It is called node-disjoint if any node except for o and d appears in
at most one path. A subgraph H of G is called K-edge-survivable (respectively, K-node-survivable)
if for any o, d ∈ V , H contains at least a specified number K of edge-disjoint (respectively, node-
disjoint) od-paths. Then, the K-edge-survivable network design problem, denoted by ESNDP ,
consists of finding a K-edge-survivable subgraph of G with minimum total cost, where the cost
of a subgraph is the sum of the costs of its edges. Similarly, the node-survivable network-design
problem, denoted by NSNDP , consists of finding a minimum-cost node-survivable subgraph of G.
A polyhedral study of the problem for the K-ESNDP with K = 2 can be found in Stoer (1993)
while the node variant is, among others, addressed in Gr¨otschel et al. (1992) and reviewed in Fortz
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(2000). The reader is also referred to Raghavan (1995) for a discussion on flow-based models and
projection from flow based models to original arc variables and to Magnanti and Raghavan (2005)
for enhancements on standard flow based models.
In general, the survivability constraints alone may not be sufficient to guarantee a cost effective
routing with a good quality of service. The reason for this is that the routing paths may be too
“long”, leading to unacceptable delays. Since in most of the routing technologies, delay is caused at
the nodes, it is usual to measure the delay in a path in terms of its number of intermediate nodes,
or equivalently, its number of arcs (or hops). Thus, to guarantee the required quality of service,
we impose a limit on the number of arcs of the routing paths. Hop-constraints were considered
by Balakrishnan and Altinkemer (1992) as a means of generating alternative base solutions for a
network design problem. Later on, Gouveia (1998) presented a layered network flow reformulation
whose linear programming bound proved to be quite tight. This reformulation has, then, been used
in several network design problems with hop-constraints (e.g, Pirkul and Soni (2003), Gouveia and
Magnanti (2003) and Gouveia et al. (2003)) and even some hop-constrained problems involving
survivability considerations (more on this below). It is also interesting to point out that the
apparently simple general network design problem with L = 2 already contains a complex structure
(see Dahl and Johannessen (2004) who also conduct a computational study of this variation of the
problem).
In this paper, we study a problem which incorporates the two requirements, survivability and
quality of service. More precisely, given a graph G and two parameters K and L, we consider an
extension of the ESNDP where each path is constrained to have at most L arcs. We note that this
is not the first time that the two types of constraints are considered together. As far as we know,
the earliest work that combined hop-constraints with the constraint that the required paths must
be node-disjoint is the bounded ring network design problem studied by Fortz and Labb´e (2002,
2004); Fortz et al. (2006), among others. The problem we study was first studied by Huygens et al.
(2007) who only consider L ≤ 4 and K = 2. The node-disjoint variant was studied by Gouveia
et al. (2006) and later in Gouveia et al. (2008) who consider a more complicated version. The
reader is referred to the survey by Kerivin and Mahjoub (2005) who consider the disjoint path case
alone, network design problems only with hop constraints and the case where the two requirements
are considered together.
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1.2 Model and method motivation
Relevant for obtaining the good computational results for the K-ESNDP is the fact that most
of the best methods rely on so-called natural models, that is, models that use only one variable
for each edge of the graph and an exponential sized set of constraints. However, for many of
these inequalities the associated separation problem is well solved and thus, they can be efficiently
separated leading to quite good cutting plane algorithms as shown for instance in Dahl and Stoer
(1998). Unfortunately, finding a similar approach for the same type of problem with hop-constraints
is much more complicated. The reason is that it is not straightforward to obtain a valid natural
formulation for the particular case of finding a set of edges containing K edge-disjoint L-paths
between the two given nodes. Ita´ı et al. (1982) and later Bley (1997) study the complexity of this
problem for the node-disjoint and the edge-disjoint cases. Recently, Bley and Neto (2010) also
studied the approximability of the problem for L = 3 and L = 4.
For K = 1, Dahl (1999) has provided such a formulation and shown that it describes the
corresponding convex hull for L ≤ 3. Later on, Dahl et al. (2004) have shown that finding such
a description for L ≥ 4 would be much more complicated. For K ≥ 2 the results are even worse.
Huygens et al. (2004) have extended Dahl’s result for K = 2 and L ≤ 3. For L ≥ 4, the only
interesting result for the moment is the one given in Huygens and Mahjoub (2007) for L = 4 and
K = 2 where a valid formulation has been given. However, in terms of valid inequalities and with
exception to the well known L-path cut inequalities, nothing is known for larger values of L. This
may also explain why the only cutting plane method for the more general problem with several
sources and several destinations by Huygens et al. (2007) only considers L ≤ 3.
Based on this history, it makes sense to look for alternative ways of formulating the problem.
The layered approach described previously appears to be a good candidate for formulating the
problem since it is easily generalized for the case with K disjoint paths. Furthermore, a similar
approach has already been used for the version of the problem with node disjoint paths (see Gouveia
et al. (2006) and Gouveia et al. (2008) for a more complicated version) and the results in these
papers (although for a slightly more complicated variant) give a sort of motivation for the method
developed and tested in this paper:
(i) the linear programming bound given by the model (if it can be solved) is often very good;
(ii) however, the model is difficult to use in a straightforward way with CPLEX because it has
too many variables.
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