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A branch-and-cut algorithm for two-level survivable network design problems

01 Mar 2016-Computers & Operations Research (Elsevier Science Ltd.)-Vol. 67, pp 102-112

TL;DR: This paper approaches the problem of designing a two-level network protected against single-edge failures by giving integer programming formulations and valid inequalities for the different versions of the problem, solving them using a branch-and-cut algorithm, and discussing computational results.

AbstractThis paper approaches the problem of designing a two-level network protected against single-edge failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs, the connection of the hubs through a backbone network (first network level), and the assignment of terminals to hubs and their connection through access networks (second network level). We consider two survivable structures in both network levels. One structure is a two-edge connected network, and the other structure is a ring. There is a limit on the number of nodes in each access network, and there are fixed costs associated with the hubs and the access and backbone links. The aim of the problem is to minimize the total cost. We give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results. Some of the new inequalities can be used also to solve other problems in the literature, like the plant cycle location problem and the hub location routing problem. HighlightsWe study two-level survivable network design problems.Two types of survivable structures are considered: rings and 2-edge connected graphs.We give an ILP formulation that models the different problem' variants.We derive strong valid inequalities and devise separation procedures for them.Some of the valid inequalities generalize other inequalities in the literature.The proposed branch-and-cut method succeeds in solving instances with up to 40 nodes.

Topics: Backbone network (60%), Network planning and design (56%), Access network (54%), Branch and cut (54%), Integer programming (52%)

Summary (3 min read)

1. Introduction

  • The authors study several two-level network design problems with survivability requirements in both levels.
  • The lower level networks are called access networks and they connect the users to hubs.
  • There are also studies on two-level networks with survivability requirements on the backbone network.
  • They analyze worst-case performances of some heuristics for some special cases.

2. MIP models

  • Note that this is not a restrictive assumption as one can solve the problem for different root nodes if one has not been a priori fixed.
  • If an edge is used in the backbone network then, due to constraints (5) and (6), both endpoints should be hubs.
  • Finally, constraints (10) and (11) are variable restrictions.
  • In particular, to model the 2EC/ring and ring/ring design problems the authors add the degree constraints xðδðiÞÞr2 8 iAV : ð12Þ.
  • These constraints limit to at most two the degree of a node in an access network and, together with the connectivity constraints (8), force those networks to be rings.

3. Valid inequalities

  • This section presents several families of valid inequalities for the four variants of the 2-LSNDP.
  • The feasible set of this problem is a relaxation of the 2EC/2EC network design problem obtained by dropping the constraints related with xe variables.
  • The authors now present three new families of inequalities.
  • There are at least rðSÞ PiA Szii distinct hubs in set V⧹S that serve nodes in S. Hence, overall at least 2 rðSÞ PiA SziiþPmk ¼ 1 zðjk : SkÞ access edges cross the cut δðSÞ.□ ð21Þ to solve the hub location routing problem.

4. Branch-and-cut algorithm

  • The authors propose an exact branch-and-cut algorithm to solve the 2- LSNDP based on the MIP model strengthened with the valid inequalities presented in the previous section.
  • The branch-and-cut approach consists of a cutting plane technique embedded into a branch-and-bound framework.
  • The authors describe next the main features of the algorithm.

4.1. Initialization

  • To start the optimization the authors solve the linear program (LP) given by the constraints (2)–(6) and variable bounds.
  • To these inequalities, the authors add the degree inequalities (12) when the access networks are required to be rings, i.e., when the variant to solve is the 2EC/ring or the ring/ring.
  • Similarly, to solve the design problems with required ring structure in the backbone network (i.e., ring/ 2EC and ring/ring) the authors add constraints (13).
  • When the backbone network does not need to be a ring, the authors replace (13) by (7) with S¼ fig for all iAV in the initial LP.

4.2. Cutting plane phase

  • If the optimal solution of the LP relaxation is integer, the authors check whether it is a feasible solution for the 2-LSNDP by applying the separation routines for constraints (14), (7) and (8).
  • The cutting plane phase is performed only each 10 branch-and-cut nodes.
  • The left hand side of all these inequalities is the same and they do not involve variables related to the backbone network.
  • The authors refer to the variable values of the current fractional solution by ðxn; yn; znÞ.

4.2.1. Separation of inequalities (7)

  • To separate constraints (7) the authors use an exact and polynomial procedure presented in [20], and inspired by the known separation algorithm for the subtour elimination constraints for the travelling salesman problem.
  • The procedure consists of solving max-fow/min-cut problems on an appropriately defined support graph.
  • Next the authors determine a min-cut set S V 0 with iAS and 0=2S, and finally they check the violation of inequality (7) for that set.

4.2.3. Separation of inequalities (14)

  • Inequalities (14) can be separated in polynomial time.
  • If the inequality for this choice of S is not violated, then there exists no violated inequality (14) for edge fi; jg.

4.2.4. Separation of inequalities (15)

  • Constraints (15) are heuristically separated using a procedure presented in [11] that works as follows.
  • The authors look for sets of nodes fv1;…; vpg that determine odd cycles in G0.
  • The set F is formed by taking an odd number of edges from δðV0Þ with fractional values yne41=2.
  • Then, the corresponding inequality is checked for violation.

4.2.5. Separation of inequalities (16)

  • The bridge inequalities (16) can be separated exactly in polynomial time with the following algorithm.
  • If the capacity of this cut is less than 2yne , the corresponding inequality (16) is violated.
  • Therefore the authors use the following strategy to reduce the number of min-cut computations.
  • Note that, with these changes, the separation procedure for inequalities (16) becomes heuristic.

4.2.6. Separation of inequalities (18)

  • The capacity of the edges in E0 is set to the positive value considered for their definition.
  • Therefore, again the separation problem can be solved exactly by performing a min-cut computation for each pair of nodes.

4.2.7. Separation of inequalities (22)

  • To separate constraints (22) the authors use a heuristic procedure based on the separation algorithm for (20) described in [27].
  • The edges in E0 are the edges in E, with capacity xne ; the edges connecting s with each iAV , with capacity 2ðznjiþ1=qÞ; and the edges connecting each iAV with t, with capacity 2znii.

4.2.8. Separation of inequalities (24)

  • This section describes a heuristic separation for (24) based on an exact and polynomial-time separation for a closely related family of inequalities.
  • These inequalities are the ones obtained by not considering the rounding up operator in (24), i.e. yðEðSÞÞþyðδðSÞÞþxðδðSÞÞZ.
  • This separation algorithm can be adapted to deal with other inequalities (23) with m¼1, like for example when r(S) is defined as in (21).

4.3. Branching strategy

  • The branching strategy the authors devised prioritizes the set of variables zjj.
  • This is done because fixing the set of hubs contributes to determine other features of the solution.
  • Branching on the other variables is done only when all variables zjj are integral.

5. Computational results

  • The authors used CPLEX 12.5 as mixed integer linear programming solver.
  • For a given number of nodes n and fix hub cost setting f j, the hardest instances are those with tighter capacity.
  • To assess the influence of the different sets of valid inequalities proposed in this work, the authors compare four versions of the branchand-cut algorithm that differ by the sets of valid inequalities considered.
  • The addition of constraints (18) reduces the gaps and the running times, this being more evident for Class II instances.
  • This happens because the violation of constraints (17) is checked inside the separation routines for (8) and (18), and the authors are certain to find a violated constraint of the first type when any of the two latter is violated.

6. Conclusions

  • This paper addresses a two-level survivable network design problem in which location, assignment, and network design tasks are jointly tackled.
  • The authors present an integer programming formulation and valid inequalities for the problem.
  • This formulation models, with minor modifications, the four possible combinations of the two singleedge protected topologies considered (rings and two-edge connected networks).
  • The authors have designed an exact branch- and-cut algorithm and they have tested its performance on two sets of instances with different degree of difficulty.
  • The results are satisfactory since they show that instances with up to 40 nodes, which is already a large size for this kind of problems, are solved to optimality in a reasonable amount of time.

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Content maybe subject to copyright    Report

A branch-and-cut algorithm for two-level survivable network
design problems
Inmaculada Rodríguez-Martín
a,
n
, Juan-José Salazar-González
a
, Hande Yaman
b
a
DMEIO, Facultad de Ciencias, Universidad de La Laguna, Tenerife, Spain
b
Department of Industrial Engineering, Bilkent University, Ankara, Turkey
article info
Available online 28 September 2015
Keywords:
Network design
Survivability
Hierarchical networks
Valid inequalities
Branch-and-cut
abstract
This paper approaches the problem of designing a two-level network protected against single-edge
failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs,
the connection of the hubs through a backbone network (rst network level), and the assignment of
terminals to hubs and their connection through access networks (second network level). We consider
two survivable structures in both network levels. One structure is a two-edge connected network, and
the other structure is a ring. There is a limit on the number of nodes in each access network, and there
are xed costs associated with the hubs and the access and backbone links. The aim of the problem is to
minimize the total cost. We give integer programming formulations and valid inequalities for the dif-
ferent versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational
results. Some of the new inequalities can be used also to solve other problems in the literature, like the
plant cycle location problem and the hub location routing problem.
& 2015 Elsevier Ltd. All rights reserved.
1. Introduction
In this paper, we study several two-level network design pro-
blems with survivability requirements in both levels. Tele-
communication networks are usually multilayer hierarchical net-
works where the trafc from different origins are collected and
sent to upper levels to be routed towards their destinations. In a
typical two-level network, the upper level is called the backbone
network, and connects the hubs (concentrators, switches, multi-
plexers) among themselves. The lower level networks are called
access networks and they connect the users to hubs. Klincewicz
[18] uses the notation backbone structure/access structure to
specify the structure of a two-level network. For instance in a
fully connected/ring network, the backbone network is a com-
plete graph between the hubs, and the access networks are rings,
each visiting a subset of users and one hub.
Network survivability, which is the ability of a network to
continue functioning in the case of failures, is one of the most
critical issues in the design of telecommunications networks. A
common assumption is that at most one edge can fail at a time in a
network. To ensure survivability in case of single edge failures, the
most common topology used is a ring. A ring is a special case of a
2-edge connected subgraph where each node has degree two. A 2-
edge connected subgraph (2EC) provides the same level of survi-
vability as a ring in case of edge failures, and may result in less
redundant capacity reservation (see, e.g., Karaşan et al. [15] and
Shi and Fonseka [29] ). We consider these two topologies in the
design of a two-level network with protection against a single
edge failure. As a result, we study the design problems associated
with four different networks: 2EC/2EC, 2EC/ring, ring/2EC and
ring/ring networks. We will denote all these problems with the
general term of 2-level survivable network design problem (2-
LSNDP).
In a 2-LSNDP we are given a set of nodes. The cost of con-
necting a pair of nodes by a link in the backbone or in an access
network is known. There is also a cost associated to select a node
as hub. The number of nodes in each access network is limited by
the capacity of the hubs, which is a priori given. The problem
consists of choosing the nodes to act as hubs and connecting them
through a backbone network, and of assigning the non-hub nodes
to the hubs and connecting them through access networks,
respecting the capacity and the topology requirements. The
objective is to minimize the total cost of the resulting two-level
network. Fig. 1 shows a ring/ring 2-LSNDP optimal solution for an
instance with 15 nodes where the number of nodes in each access
network is limited to 3. The solid lines represent the backbone
network and the dashed lines represent the access networks. The
nodes in the backbone network are the hubs. Fig. 2 shows the
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/caor
Computers & Operations Research
http://dx.doi.org/10.1016/j.cor.2015.09.008
0305-0548/& 2015 Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail addresses: irguez@ull.es (I. Rodríguez-Martín),
jjsalaza@ull.es (J.-J. Salazar-González), hyaman@bilkent.edu.tr (H. Yaman).
Computers & Operations Research 67 (2016) 102112

optimal solution for the same instance when the required network
structure is 2EC/2EC.
Even though survivability is critical for service providers, there
are few studies on designing hierarchical survivable networks.
Most studies on survivable network design problems consider a
single layer of the network. For reviews of these studies, one can
refer to Grötschel et al. [13] and Kerivin and Mahjoub [16]. Poly-
nomially solvable special cases of the survivable network design
problem are studied in Kerivin and Mahjoub [17]. The most
common network structure in this eld is a 2EC subgraph (see,
e.g., [23,24,31,33]). Problems related with designing rings of
bounded size are studied by Fortz and Labbé [7] and Fortz et al. [8
10]. Generalizations of 2EC networks are studied by Magnanti and
Raghavan [22] and Balakrishnan et al. [2].
There are also studies on two-level networks with survivability
requirements on the backbone network. For example, Labbé et al.
[20] propose a branch-and-cut algorithm for designing a ring as
backbone network, while the access networks are direct
connections from users to a hub (i.e., star structure). Baldacci et al.
[6] address a more general problem where the backbone network
allows m rings instead of a single one. Fouilhoux et al. [11] study
the variant where the ring structure is replaced by a 2EC network.
In all these studies the access networks are forced to be star
structures.
The studies that consider survivability at both layers of the
network are few. Lee and Koh [21] study the ring/chain network
design problem with dual homing where the ring topology in the
backbone network is given. They study the design of the access
networks. They show that the problem is NP-hard, propose an
integer programming formulation and describe a tabu search
heuristic. Thomadsen and Stidsen [32] study the ring/ring network
design problem. They suggest to solve the design problems asso-
ciated with different levels sequentially. They propose a branch-
and-price algorithm for this purpose. Carroll and McGarraghy [3]
also propose to decompose the problems of designing the rings in
different levels. Shi and Fonseka [28] study the design of hier-
archical self healing rings and propose a heuristic. Proestki and
Sinclair [26] and Shi and Fonseka [29] propose heuristic algo-
rithms for the problem with dual homing. Balakrishnan et al. [1]
study a generalization of the two-level survivable network design
problem to that of multitiers. They analyze worst-case perfor-
mances of some heuristics for some special cases. Park et al. [25]
study a node clustering problem with survivability requirements.
Karasan et al. [15] propose a branch-and-cut algorithm for the
problem where the backbone network is 2EC and each user node
is connected directly to two distinct hub nodes. More recently, Hill
and Voß [14] introduce the capacitated ring tree problem where
nodes are connected with rings and trees, and rings intersect at a
distributor node.
For further related studies, we refer the readers to the follow-
ing surveys. Klincewicz [18] reviews design problems that involve
location of hubs. Gourdin et al. [12] survey location problems
encountered in telecommunications network design. Soriano et al.
[30] provide an overview of design and dimensioning problems in
survivable SDH/SONET networks. Studies on combined location
and network design problems are reviewed by Contreras and
Fernández [4].
In summary, there are few studies that consider the design of
two-level networks with survivability requirements in both levels.
Most of such studies are on designing ring/ring networks and most
of the proposed approaches are of heuristic nature. The con-
tribution of this paper is to propose strong formulations and exact
solution methods for the two-level survivable network design
problem where both rings and 2-edge connected networks are
used to ensure survivability.
The remainder of the paper is organized as follows. The
mathematical model for the different variants of the problem is
given in Section 2. Section 3 presents several families of valid
inequalities to strengthen the linear-programming relaxations.
Section 4 details a branch-and-cut approach based on the for-
mulation and inequalities presented in the previous sections. The
performance of this approach is analyzed in Section 5 on a large
collection of instances. Finally, the paper ends with conclusions in
Section 6.
2. MIP models
We rst introduce the notations. Let V ¼f0; 1; ; n 1g be the
set of nodes, where node 0 stands for the root and is considered to
be a hub. Note that this is not a restrictive assumption as one can
solve the problem for different root nodes if one has not been a
priori xed. Let E ¼ffi; jg : i; jA V; io jg be the set of potential links.
We assume G ¼ðV; EÞ to be an undirected graph and we do not
Fig. 1. A ring/ring solution example.
Fig. 2. A 2EC/2EC solution example.
I. Rodríguez-Martín et al. / Computers & Operations Research 67 (2016) 102112 103

allow multiple edges. Installing a hub at node jA V has a cost f
j
. For
each edge eA E, the cost of installing a backbone link or an access
link on e is denoted by b
e
and a
e
, respectively. For SD V, let δðSÞ be
the set of edges with one endpoint in S, and let E(S) be the set of
edges with both endpoints in set S. When S is a singleton, i.e.,
S ¼fig, we use
δðiÞ for δðfi.
The aim of the 2-LSNDP is to partition the set of nodes V into
disjoint subsets, each with at most q nodes, choose one node from
each subset to locate a hub, and connect the hubs and the subsets
with survivable networks at minimum cost. The survivable net-
work may be either a 2EC structure or a ring, depending on the
problem variant. For simplicity in notation, rings or 2EC structures
are assumed to involve at least three nodes. The four variants of
the problem can be described with the same mathematical vari-
ables as follows. Let us dene z
ij
to be 1 if node iA V is assigned to
hub jA V, and to be 0 otherwise. Node j is a hub when z
jj
is 1. In
addition, we dene x
e
to be 1 if edge eA E is used in an access
network and 0 otherwise, and y
e
to be 1 if edge e is used in the
backbone network, and 0 otherwise. For brevity in notation, we
write xðE
0
Þ¼
P
e A E
0
x
e
and yðE
0
Þ¼
P
e A E
0
y
e
for all E
0
D E, and zðS : TÞ
instead of
P
i A S;j A T
z
ij
for all S; T D V.
The 2EC/2EC design problem, where the backbone and access
networks are required to be 2-edge connected, can be modeled as
follows:
min
X
i A V
f
i
z
ii
þ
X
e A E
a
e
x
e
þ
X
e A E
b
e
y
e
ð1Þ
s:t: zði : VÞ¼1 8iA V; ð2Þ
zðV : iÞr qz
ii
8iA V; ð3Þ
z
00
¼ 1; ð4Þ
z
ij
þy
fi;jg
r z
jj
8fi; jgA E; ð5Þ
z
ji
þy
fi;jg r z
ii
8fi;jg A E;
ð6Þ
yð
δðSÞÞ Z 2 zði : SÞ8SD Vf0g; iA S; ð7Þ
xð
δðSÞÞ Z 2zði : VSÞ8S V; iA S; ð8Þ
x
fi;jg
þz
ii
0
þz
jj
0
r 2 8fi; jgA E; i
0
; j
0
A V : i
0
a j
0
; ð9Þ
x
e
; y
e
A f0; 1g8eA E; ð10Þ
z
ij
A f0; 1g8i; jA V: ð11Þ
The objective function (1) is the sum of the cost of locating
hubs and the cost of installing access and backbone links. Con-
straints (2) ensure that each node is either a hub or it is assigned to
another node. Constraints (3) are capacity constraints that limit to
q the number of nodes assigned to a hub. They also ensure that no
node is assigned to a non-hub node. Constraint (4) forces the root
node to be a hub. If an edge is used in the backbone network then,
due to constraints (5) and (6), both endpoints should be hubs.
Otherwise, one endpoint can be assigned to the other only if the
latter is a hub. Constraints (7) impose 2-edge connectedness of the
backbone network. If node i is a hub or if it is assigned to a hub
node in set S, then there exists at least one hub in set S and the
constraint asks for at least two backbone edges on the cut
δðSÞ
since the root is in VS. Similarly, constraints (8) ensure 2-edge
connectedness of the access networks. If node iA S is allocated to a
hub node in VS then there should be at least two access links
between S and VS. Constraints (9) make sure that if the access
link fi; jg is used then i and j are allocated to the same hub. Finally,
constraints (10) and (11) are variable restrictions.
The above formulation needs minor modications to model the
other 2-LSNDP variants. In particular, to model the 2EC/ring and
ring/ring design problems we add the degree constraints
xð
δðiÞÞr 2 8 iA V: ð12Þ
These constraints limit to at most two the degree of a node in an
access network and, together with the connectivity constraints (8),
force those networks to be rings. Note that we cannot use an
equation in (12) because not all nodes must be necessarily on an
access network.
Similarly, to model the ring/2EC and ring/ring design problems
we add the degree equations
yð
δðiÞÞ ¼ 2z
ii
8iA V; ð13Þ
to ensure that each hub node i has two backbone edges adjacent to
it.
3. Valid inequalities
This section presents several families of valid inequalities for
the four variants of the 2-LSNDP. We rst give valid inequalities
that are adopted from the literature. Validity proofs are omitted as
they are very similar to the ones that appear in the cited refer-
ences. Later the section presents theorems with new inequalities,
some of which generalize other inequalities for different problems
in the literature.
Rodríguez-Martín et al. [27] study the hub location routing
problem. Several families of valid inequalities for that problem are
also valid for our problems. In particular, let ½i; jA E and S V such
that i A S and jA VS. The inequality
x
fi;jg
r zði : VSÞþzðj : SÞð14Þ
is valid for 2-LSNDP. The inequality says that if node i is assigned to
a hub in S and j to a hub in VS, i.e., if zði : VSÞ¼0 and zðj : SÞ¼0,
then as they are in separate access networks, edge fi; jg cannot be
used as an access edge. Note that these inequalities are stronger
than constraints (9).
Fouilhoux et al. [11] derive F-partition inequalities for the
network design problem where the backbone is 2EC and the access
networks are stars. The feasible set of this problem is a relaxation
of the 2EC/2EC network design problem obtained by dropping the
constraints related with x
e
variables. Hence the valid inequalities
proposed by Fouilhoux et al. [11] are also valid for our problems.
Let V
0
; ; V
p
be a partition of V such that V
l
a , for l¼0,,p and
0A V
0
. Let i
l
A V
l
for l¼1,,p and F D δðV
0
Þ such that j F 2kþ1
for some integer kZ 0. Let
δðV
0
; ; V
p
Þ be the set of edges whose
endpoints are in different sets of the partition. The inequality
yð
δðV
0
; ; V
p
ÞFÞþ
X
p
l ¼ 1
zði
l
: VV
l
ÞZ p k ð15Þ
is called F-partition and it is valid for 2-L SNDP.
Baïou and Mahjoub [5] use the following constraints to avoid
the occurrence of bridges, i.e., of cuts of cardinality one:
yðδðSÞÞ Z 2y
e
8S V; eA δðSÞ; ð16Þ
which are valid for 2-LSNDP. Clearly a network is protected against
the failure of an edge when no edge is a bridge.
We now present three new families of inequalities. The rst
family extends the classical subtour elimination constraints.
Theorem 1. Let S V be a non-empty set. Let ðS
1
; ; S
m
1
Þ be a
partition of S and ðT
1
; ; T
m
2
Þ be a partition of VS. Consider i
1
; ;
i
m
2
distinct nodes in S and j
1
; ; j
m
1
distinct nodes in VS. Then the
I. Rodríguez-Martín et al. / Computers & Operations Research 67 (2016) 102112104

inequality
xð
δðSÞÞ Z 2
X
m
2
k ¼ 1
zði
k
: T
k
Þþ
X
m
1
l ¼ 1
zðj
l
: S
l
Þ
!
ð17Þ
is valid for 2-LSNDP.
Proof. If zði
k
: T
k
Þ¼0 and zðj
l
: S
l
Þ¼0 for all k and l then the
constraint vanishes. Otherwise, each i
k
assigned to a hub in T
k
and
each j
l
assigned to a hub in S
l
involves a different access network,
which implies that at least two access edges cross the cut
δðSÞ
for each.
Inequalities (17) generalize the following inequalities intro-
duced by Labbé et al. [19] for the plant cycle location problem:
xð
δðSÞÞ Z 2 zði : VSÞþzðj : SÞ

8S V; iA S; jA VS: ð18Þ
Note that inequalities (18) dominate constraints (8).
The second family of inequalities exploits the capacity limita-
tion, as in the classical vehicle routing problem.
Theorem 2. Consider S V, a partition ðS
1
; ; S
m
Þ of S and distinct
nodes j
1
; ; j
m
be in VS. Let r(S) be a lower bound on the number of
access networks serving nodes in S. Then the inequality
xð
δðSÞÞ Z 2 rðSÞ
X
i A S
z
ii
þ
X
m
k ¼ 1
zðj
k
: S
k
Þ
!
ð19Þ
is valid for 2-LSNDP.
Proof. There are at least rðS Þ
P
i A S
z
ii
distinct hubs in set VS that
serve nodes in S. There are also at least
P
m
k ¼ 1
zðj
k
: S
k
Þ distinct
hubs in S serving nodes in VS. Hence, overall at least 2
rðSÞ
P
i A S
z
ii
þ
P
m
k ¼ 1
zðj
k
: S
k
Þ

access edges cross the cut δðSÞ.
Examples of lower bounds r(S)are
j Sj
q
and
j Sxð
δ
ðSÞÞ=2
q
, though
the latter is valid only when the access networks are required to be
rings and not for the general 2-LSND. Rodríguez-Martín et al. [27]
use the inequalities
xð
δðSÞÞ Z 2
j Sj
q
X
i A S
z
ii
!
ð20Þ
xð
δðSÞÞ Z 2
j SxðδðSÞÞ=2
q
X
i A S
z
ii
!
ð21Þ
to solve the hub location routing problem. Note that inequalities
(19) dominate these inequalities.
A particular case of inequalities (19) are
xð
δðSÞÞ Z 2
j Sj
q
X
i A S
z
ii
þzðj : SÞ
!
8S V; jA VS: ð22Þ
Finally, the next result proposes a third family of inequalities.
These inequalities are different from previous inequalities as they
involve variables associated with both access and backbone edges.
Theorem 3. Let S D Vf0g be such that S a , let ðS
1
; ; S
m
Þ be a
partition of S, and consider j
1
; ; j
m
be distinct nodes in VS. Let r(S)
be a lower bound on the number of access networks serving nodes in
S. The inequality
yðEðSÞÞþ yð
δðSÞÞ þxðδðSÞÞ Z rðSÞþ1þ 2
X
m
k ¼ 1
zðj
k
: S
k
Þð23Þ
is valid for 2-LSNDP.
Proof. Given a feasible solution, let S
0
1
; ; S
0
p
be the partition of the
set S into access networks. We know that pZ rðSÞ since each access
network can have at most q nodes. Let i
k
be the hub serving the
access network S
0
k
, K ¼f1; ; pg, K
0
¼fkA K : i
k
A S
0
k
g and
S
0
¼[
k A K
0
S
0
k
.
If K
0
a , then yðEðSÞÞþyðδ ðSÞÞ ¼ yðEðS
0
ÞÞþ yðδðS
0
ÞÞZ j K
0
1. For
kA KK
0
, we know that there are at least two access edges
between S
0
k
and VS, i.e., xðS
0
k
: VSÞZ 2. In addition xðδðSÞÞZ
P
k
A KK
0
xðS
0
k
: V SÞ.SoyðEðSÞÞþ yðδðSÞÞþ xðδðSÞÞZ j K
0
1þ 2ðjK
K
0
.IfK
0
¼ , then since xðS
0
k
: VSÞZ 2 for k ¼ 1; ; p,wehave
yðEðSÞÞ þyð
δðSÞÞ þxðδðSÞÞ Z 2p.Bothj K
0
1þ2ðjKK
0
and 2p are
greater than or equal to pþ 1, and pþ1Z rðSÞþ1. So inequality yð
EðSÞÞ þyð
δðSÞÞ þxðδðSÞÞ Z rðSÞþ1 is satised.
Now there are at least
P
m
k ¼ 1
zðj
k
: S
k
Þ distinct nodes in VS that
are assigned to distinct hub nodes in S. Hence there are also 2
P
m
k ¼ 1
zðj
k
: S
k
Þ access edges crossing the cut.
A particular case of inequality (23) is
yðEðSÞÞ þyð
δðSÞÞ þxðδðSÞÞ Z
j Sj
q
þ1þ2zðj : SÞð24Þ
where jA VS. When S a is singleton, inequality (24) becomes
xð
δðiÞÞþyðδ ðiÞÞZ 2þ 2z
ji
ð25Þ
where i; jA V and ia j.
4. Branch-and-cut algorithm
We propose an exact branch-and-cut algorithm to solve the 2-
LSNDP based on the MIP model strengthened with the valid
inequalities presented in the previous section. The branch-and-cut
approach consists of a cutting plane technique embedded into a
branch-and-bound framework. We describe next the main fea-
tures of the algorithm.
4.1. Initialization
To start the optimization we solve the linear program (LP) given
by the constraints (2)(6) and variable bounds. To these inequal-
ities, we add the degree inequalities (12) when the access net-
works are required to be rings, i.e., when the variant to solve is the
2EC/ring or the ring/ring. Similarly, to solve the design problems
with required ring structure in the backbone network (i.e., ring/
2EC and ring/ring) we add constraints (13). When the backbone
network does not need to be a ring, we replace (13) by (7) with
S ¼fig for all iA V in the initial LP.
4.2. Cutting plane phase
If the optimal solution of the LP relaxation is integer, we check
whether it is a feasible solution for the 2-LSNDP by applying the
separation routines for constraints (14), (7) and (8). Otherwise, we
apply the separation procedures for constraints (14), (25), (7), (8),
(24), (22), (16), (15), and (18), in this sequence. The separation
procedure for the last family of constraints is applied only if no
other violated cuts have been found, due to its computational cost.
The cutting plane phase is performed only each 10 branch-and-cut
nodes. Moreover, the number of violated cuts of each family added
to the model is limited to 15, and the total number of cuts added in
each cut generation step is limited to 75.
Constraints (25) are separated in Oð n
2
Þ by complete enumera-
tion. To separate constraints (7), (8), (14), and (18), we follow
approaches presented in [20] and [27]. We devised separation
methods for constraints (16), (22), and (24). To separate the F-
partition inequalities (15) we use a heuristic algorithm given in
[11].
Finally, constraints (17) are not directly separated, but their
violation is checked each time a violated inequality (8) or (18) is
I. Rodríguez-Martín et al. / Computers & Operations Research 67 (2016) 102112 105

found. The left hand side of all these inequalities is the same and
they do not involve variables related to the backbone network. The
violation test for (17) is done heuristically by checking whether
xð
δðSÞÞ Z 2
X
k A VS
max
i A S
zði : kÞþ
X
l A S
max
j A VS
zðj : lÞ
0
@
1
A
:
We next outline the separation procedures we have imple-
mented. We refer to the variable values of the current fractional
solution by ðx
n
; y
n
; z
n
Þ.
4.2.1. Separation of inequalities (7)
To separate constraints (7) we use an exact and polynomial
procedure presented in [20], and inspired by the known separa-
tion algorithm for the subtour elimination constraints for the
travelling salesman problem. The procedure consists of solving
max-fow/min-cut problems on an appropriately dened support
graph. Note that inequalities (7) can be written as
yð
δðSÞÞ þ2 zði : VSÞZ 2 8 SD Vf0g; iA S :
For each node iA Vf0g,wedene a support graph G
0
¼ðV
0
; E
0
Þ
with V
0
¼ V and E
0
¼ E. The capacity of edges fi; jg A E with ja i is
set to y
n
ij
þ2z
n
ij
, and all other edges eA E are assigned a capacity
equal to y
n
e
. Next we determine a min-cut set S V
0
with iA S and
0=2S, and nally we check the violation of inequality (7) for
that set.
4.2.2. Separation of inequalities (8)
Constraints (8) can be separated in polynomial time with a
procedure similar to the previous one. Just note that inequalities
(8) can be written as
xð
δðSÞÞ þ2 zði : SÞZ 2 8 S V; iA S:
Since constraints (8) are dominated by constraints (18), each
time we nd a violated inequality (8) we replace it by the con-
straint (18) dened as
xð
δðSÞÞ Z 2 zði : VSÞþzðj : SÞ

;
where j is a node in VS that maximizes z
n
ðj : SÞ.
4.2.3. Separation of inequalities (14)
Inequalities (14) can be separated in polynomial time. For a
given edge fi; jgA E,wedene S ¼fig[fkA Vfjg : z
n
ik
Z z
n
jk
g. If the
inequality for this choice of S is not violated, then there exists no
violated inequality (14) for edge fi; jg.
4.2.4. Separation of inequalities (15)
Constraints (15) are heuristically separated using a procedure
presented in [11] that works as follows. Let G
0
¼ðV
0
; E
0
Þ be a sup-
port graph with V
0
¼fiA V : z
n
ii
4 0g[fiA V : z
n
ði : Vfi4 0 and
y
n
ðδðiÞÞ 4 0g and E
0
¼ffi; jg A E : y
n
ij
4 0g. We look for sets of nodes
fv
1
; ; v
p
g that determine odd cycles in G
0
. For each such set, we
dene V
0
¼ Vfv
1
; ; v
p
g, and V
i
¼fv
i
g for i ¼ 1; ; p. The set F is
formed by taking an odd number of edges from
δðV
0
Þ with frac-
tional values y
n
e
4 1=2. Then, the corresponding inequality is
checked for violation.
4.2.5. Separation of inequalities (16)
The bridge inequalities (16) can be separated exactly in poly-
nomial time with the following algorithm. For each edge e A E with
y
n
e
4 0 we determine in the support graph G
0
¼ðV
0
; E
0
Þ with V
0
¼ V
and E
0
¼fe
0
A E : y
n
e
0
4 0g the min-cut set S separating the two nodes
associated with e. If the capacity of this cut is less than 2y
n
e
, the
corresponding inequality (16) is violated.
Although this algorithm is polynomial it might produce a large
number of similar violated inequalities, with the consequent loss
of time. Therefore we use the following strategy to reduce the
number of min-cut computations. We consider only edges e ¼fi; j
gA E with y
n
e
4 0:5 and such that i and j have not been extremes of
an edge producing a violated inequality previously. Moreover, we
associate to each set S in a violated inequality a given label, and we
check for duplicates before adding the new cut to the LP. Note that,
with these changes, the separation procedure for inequalities (16)
becomes heuristic.
4.2.6. Separation of inequalities (18)
Constraints (18) can be written as
xð
δðSÞÞ þ2 zði : SÞþ2zðj : VSÞ Z 4 8S V; iA S; jA VS:
For each pair of nodes i; jA V let us dene a support graph G
0
¼ð
V
0
; E
0
Þ where V
0
¼ V and E
0
contains all edges fi; kgA E such that
kA V and x
n
ik
þ2z
n
jk
4 0, all edges fj; kg A E such that kA V and
x
n
jk
þ2z
n
ik
4 0, and all other edges eA E such that x
n
e
4 0. The capa-
city of the edges in E
0
is set to the positive value considered for
their denition. Let S V
0
be such that iA S, j=2S, and δðSÞ is the
minimum cut between i and j in G
0
. If the capacity of δðSÞ is smaller
than 4, S denes the most violated constraint (18) for i and j.
Therefore, again the separation problem can be solved exactly by
performing a min-cut computation for each pair of nodes.
4.2.7. Separation of inequalities (22)
To separate constraints (22) we use a heuristic procedure based
on the separation algorithm for (20) described in [27]. It starts by
separating exactly and in polynomial-time the inequalities with-
out the rounding up operator, i.e.
xð
δðSÞÞ Z 2
j Sj
q
X
i A S
z
ii
þzðj : SÞ
!
with S Vf0g and jA VS. These inequalities are equivalent to
xð
δðSÞÞ þ
2j VSj
q
þ2zðj : VSÞþ
X
i A S
2z
ii
Z 2
j V j
q
þ1

:
Then, for each jA V, nding a set S de ning the most violated
inequality (if any) by a given solution ðx
n
; y
n
; z
n
Þ aims at performing
a st min-cut computation on a capacitated network G
0
¼ðV
0
; E
0
Þ
with V
0
¼ V [fsg[ftg, being s and t two dummy nodes. The edges
in E
0
are the edges in E, with capacity x
n
e
; the edges connecting s
with each i A V, with capacity 2ðz
n
ji
þ1=qÞ; and the edges connect-
ing each iA V with t, with capacity 2z
n
ii
. To guarantee that j=2S we
must increase the capacity of the edge fj; tg by adding a large
amount. We nally check the potential violation of the inequality
(22) dened by the set S in the side of s generated by the st min-
cut computation.
4.2.8. Separation of inequalities (24)
This section describes a heuristic separation for (24) based on
an exact and polynomial-time separation for a closely related
family of inequalities. These inequalities are the ones obtained by
not considering the rounding up operator in (24), i.e.
yðEðSÞÞ þyð
δðSÞÞ þxðδðSÞÞ Z
j Sj
q
þ1þ2zðj : SÞ;
with S Vf0g and jA VS. These inequalities are equivalent to
yðEðSÞÞ þ
yðδðSÞÞ
2
j Sj
q
þxð
δðSÞÞ þ
yðδðSÞÞ
2
þ2zðj : VSÞZ 3;
and the rst three terms in the left-hand side can be written as
yðEðSÞÞ þ
yðδðSÞÞ
2
j Sj
q
¼
X
i A S
yðδðiÞÞ
2
1
q

¼
X
i A S:
yð
δ
ðiÞÞ 4 2=q
yðδðiÞÞ
2
1
q

I. Rodríguez-Martín et al. / Computers & Operations Research 67 (2016) 102112106

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Frequently Asked Questions (1)
Q1. What are the contributions in "A branch-and-cut algorithm for two-level survivable network design problems" ?

This paper approaches the problem of designing a two-level network protected against single-edge failures. The authors consider two survivable structures in both network levels. The aim of the problem is to minimize the total cost. The authors give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results.