A Cutting Plane Algorithm forMulticommodity Survivable Network DesignProblems
Geir Dahl
∗
and Mechthild Stoer
†
April, 1996
Abstract
We present a cutting plane algorithm for solving the followingtelecommunicationsnetworkdesign problem:
given
pointtopoint trafﬁc demands in a network, speciﬁed survivability requirements and adiscrete cost/capacity function for each link,
ﬁnd
minimum cost capacity expansions satisfying the given demands. This algorithm is basedon the polyhedral study described in [19]. In this paper we describethe underlying problem, the model and the main ingredients in ouralgorithm. This includes: initial formulation, feasibility test, separation for strong cutting planes and primal heuristics. Computationalresults for a set of realworld problems are reported.
1 Introduction
The design of costeﬃcient telecommunications networks meeting requirements concerning traﬃc, survivability etc. is a major challenge with greateconomic impact. In particular, it is important to establish networks thatare robust with respect to accidents like cable cuts, electronic failures orpower supply shutdown. The disastrous consequences of a ﬁre in switching center in Chicago 1988 are described in [21]. Lost business was thenestimated to be hundreds of millions of dollars. [7] describes the social consequences of a ﬁre in a switching center in Oslo in 1985, which interrupted
∗
Institute of Informatics, University of Oslo, P.O.Box 1080 Blindern, 0316 Oslo,Norway. Email:geird@iﬁ.uio.no
†
Telenor Research, P.O.Box 83, 2007 Kjeller, Norway. Email: stoer@nta.no
1
28.000 subscriber lines. (Hardest hit were elder and sick people, and businesses with many clients and suppliers.) Such failures happen more often,but qualiﬁed estimates of their consequences are seldom published.Often, the capacity limitations play a crucial role in these design problems(e.g., capacities of terminal equipment installed at both end nodes of eachtransmission links). This calls for models and methods for designing lowcost capacitated networks that allow routing of traﬃc in both normal andspeciﬁed failure situations. The model we discuss in this paper falls into thisframework.In MULTISUN (MULTIcommodity SUrvivable Network design) we integrate the problems of topological design, capacity assignment and routing.Due to its generality this is a very diﬃcult problem with
NP
hard problemsas special cases. The main purpose of this paper is to describe a cuttingplane algorithm for solving MULTISUN problems and report computationalresults for some realworld problems of interest.In an earlier paper [19] we presented a theoretical study of the MULTISUN problem and identiﬁed several classes of facet deﬁning inequalities forcertain associated integral polyhedra. We therefore refer to [19] for validityand facet proofs for the inequalities discussed later.The MULTISUN problem can be described more precisely as follows.Let
V
be a given set of nodes with traﬃc demands between certain pairsof these nodes. Each demand represents a certain amount of pointtopointtraﬃc to be routed in the network between the srcin and destination nodes.Traﬃc may be split on several paths, so it may be viewed as a continuousnetwork ﬂow. In addition, we have given “supply” edges joining pairs of nodes in
V
; these represent existing or potential direct physical links (e.g.,a ﬁber cable or a radio relay system). For each supply edge one wants todecide which capacity to install, selected from a discrete set of alternatives,each with an associated building cost. We are interested in costoptimalcapacity extensions that satisfy the following conditions:(i) in case of a node or edge failure, all demands can be routed simultaneously,(ii) when all nodes and edges are operating, all demands can be routedsimultaneously such that no more than a given fraction of the givendemand is routed through any intermediate node.This model can be viewed as an extension of the work of Gr¨otschel et al.[8] on the design of networks with connectivity constraints. There each edge,once installed, was supposed to have inﬁnite capacity. This leads typicallyto very sparse designs. In this study, we combine connectivity with demand2
routing to obtain more realistic designs. A large amount of work has beendone by Minoux and others on a model related to ours, with a
continuous
cost function, see Minoux [15] (and its references). Gavish et al. [6]investigate exactly the same problem as we do (without diversiﬁed routing),but they use another LPformulation. They experienced diﬃculties with thenumber of routing variables, which are projected away in our model. Lee etal. [10] investigate spare channel assignment for upgrading an existing network with two types of link facilities under survivability constraints. The“routings” in the nonfailure state are here assumed to be given, so only aﬂow, not a multiﬂow, needs to be found in each failure state. Lisser et al. [12]describe a promising interior point algorithm for designing a reserve network(routings in the nonfailure state are assumed to be given) with linear costson capacity increases, and with varying strategies for rerouting demands. Incontrast to the latter two references, we attempt to design the core and thereserve network at the same time, allowing dynamic rerouting of all demandsin case of failure.This paper is organized as follows. In Section 2 we present the integer linear programming model for the MULTISUN problem. In addition we explainhow to obtain stronger LP relaxations by adding certain classes of valid inequalities srcinating from knapsacklike substructure of the srcinal model.Next, in Section 3, we explain the main components of our cutting planealgorithm for the MULTISUN problem, including separation algorithms andprimal heuristics. The development of the algorithm was funded by Telenor.Results for some realworld problems are reported and discussed in Section 4.Finally, in the concluding section, some directions for further work is pointedout.We use fairly standard notation from graph theory and polyhedral theory,see [2, 17], but a few notions need to be explained.
R
M
denotes the vectorspace of real vectors indexed by
M
(where
M
is some ﬁnite set), and for
x
∈
R
M
and
S
⊆
M
we let
x
(
S
) denote
i
∈
S
x
i
. Vectors are indicated bybold face letters.
1
is a suitably dimensioned vector with 1’s. Let
G
= (
V,E
)be an undirected graph without loops and multiple edges. If
W
⊆
V
∪
E
,we let
G
−
W
denote the graph obtained from
G
by removing from
G
eachnode in
W
with their incident edges, and all edges in
W
. The
cut
δ
G
(
W
)induced by a subset
W
of
V
is the set of edges with one end node in
W
andthe other outside
W
. By
G
[
W
] = (
W,E
(
W
)) we denote the graph inducedby node set
W
, i.e.,
E
(
W
) are those edges with both end nodes in
W
. Fortwo nodes
u
and
v
, a [
u,v
]path
P
is a sequence of consecutive nodes andedges connecting
u
and
v
without repeating any nodes. A graph
G
is said tobe
2edge
(or
2node
)
connected
with respect to some given node set
R
,if between any two nodes
u,v
∈
R
there exist at least two edge (or node)3
disjoint [
u,v
]paths. In this deﬁnition
G
is not permitted to have paralleledges. A
network
N
= (
G,
c
) is a graph
G
with weights (e.g., capacitiesor demands)
c
e
≥
0 associated with each edge
e
. Given a
supply
network(
G,
c
) and a
demand
network (
H,
d
), where
G
= (
V,E
) and
H
= (
V,D
)have the same node set, a
multicommodity ﬂow
(w.r.t. (
H,
d
)) is deﬁnedas a collection of [
u,v
]paths
P
iuv
in
G
together with numbers
f
iuv
≥
0 foreach [
u,v
]
∈
D
and
i
such that
i
f
iuv
=
d
uv
, for each [
u,v
]
∈
D
. Theassociated
uv
ﬂow
is the vector
z
uv
∈
R
E
with
e
’th component given by
z
uv,e
=
i
:
e
∈
P
iuv
f
iuv
(called the
uv
ﬂow in edge
e
). The network (
G,
c
), orthe capacity vector
c
, is said to
allow
a multicommodity ﬂow w.r.t. (
H,
d
),if
uv
∈
D
z
uv,e
≤
c
e
for each
e
∈
E
, i.e., the total ﬂow in each edge does notexceed the edge capacity.
2 Mathematical model and improved formulations
In this section we ﬁrst present the mathematical model for the MULTISUNproblem and discuss multicommodity ﬂow requirements in some detail. Nextwe brieﬂy describe the polyhedral approach to this problem and how it leadsto some stronger LP formulations of the problem.
2.1 The MULTISUN model
Each edge of the
supply
graph
G
= (
V,E
) corresponds to a physical (transmission) link that has been or can be established. The nodes correspond toswitching points. We assume that
G
is connected (otherwise the problemwould decompose). In the
demand
graph
H
= (
V,D
) each edge [
u,v
]
∈
D
represents a traﬃc demand of value
d
uv
between its end nodes
u
and
v
. Foreach supply edge
e
∈
E
one has to choose a capacity
y
e
from a small setof possible choices with associated costs. This should be done so that thenetwork (
G,
y
) (with capacity vector
y
) can support the required traﬃc withtotal cost as low as possible.The possible capacity choices on each edge give rise to a discrete costfunction, whichcan be modeledas follows. For eachedge
e
letthe incrementalcapacity steps be
m
te
>
0, for
t
= 1
,...,T
e
and let the incremental cost stepsbe
c
1
≥
0, for
t
= 1
,...,T
e
. The cost of installing a capacity
y
e
at edge
e
with
s
−
1
t
=1
m
te
< y
e
≤
st
=1
m
te
is
st
=1
c
te
for
s
= 1
,...,T
e
. If an edge hasan existing capacity of
k
units, set
m
1
e
=
k
and
c
1
e
= 0. The jump in costoccuring for each capacity
st
=1
m
te
may be due to e.g. the installation of anew cable. We assume that
m
T
e
e
≥
uv
∈
D
d
uv
. If this capacity is technically4
not possible, one can associate a “high” cost
c
T
e
e
with it. This means thatall demands may be routed through any edge (but possibly at a high cost).We model the cost function using a binary variable
x
te
for each incrementalcapacity step
t
on each supply edge
e
. For each
e
the variables
x
1
e
,...,x
T
e
e
arerequired to be a (possibly empty) sequence of ones followed by a (possiblyempty) sequence of zeros; this determines the capacity range. The indexset of these (design) variables
x
te
is
I
:=
{
(
e,t
)

t
= 1
,...,T
e
,e
∈
E
}
,and
x
∈
R
I
is a
design vector
consisting of all these variables (with someordering). For a design vector
x
∈
R
I
the corresponding cost is
c
T
x
and the
associated capacity vector
y
is given by
y
e
=
T
e
t
=1
m
te
x
te
.We model the ﬂow requirements as follows. Let
y
be the capacity vectorassociated with some design vector
x
. The network (
G,
y
) is supposed toallow a multicommodity ﬂow carrying all traﬃc (in which case
y
is calledfeasible). This requirement on
y
may be expressed in terms of linear inequalities as follows. For some given nonnegative vector
µ
∈
R
E
and demand edge
f
∈
D
let
π
µf
denote the shortest path length in
G
between the two end nodesof
f
with respect to edge lengths
µ
e
,
e
∈
E
. Then
y
is feasible if and only if
e
∈
E
µ
e
y
e
≥
f
∈
D
π
µf
d
f
for all
µ
≥
0. (1)This characterization of feasible capacities is known as the “Japanese theorem” (ﬁrst stated in [9, 16]) and may be proved using linear programmingduality (some more details are found in Subsection 3.2, see also [13]). Wecall each inequality in (1) a
metric inequality
, see [19] for more commentson these inequalities.In (1) we can restrict ourselves to a
ﬁnite
set of these inequalities; namelythose deﬁned by vectors (
µ
,
π
) in the set Π of extreme rays of the cone
{
µ
∈
R
E
,
π
∈
R
D

µ
≥
0
, π
f
=
π
µf
for all
f
∈
D
}
.
(2)An important special case of the metric inequalities is obtained by choosing
µ
as the incidence vector of the
cut
δ
G
(
W
) induced by a node set
W
=
∅
,
W
=
V
(when we assume that
G
[
W
] and
G
[
V
\
W
] are connected). Then(1) reduces to the
cut inequality
y
(
δ
G
(
W
))
≥
d
(
δ
H
(
W
))
.
(3)This inequality assures that the total capacity of a cut is no smaller than allthe demands across this cut.Let
G
= (
V,E
) and
H
= (
V,D
) be as above. We model the networkfailures as follows. Consider a failing component
s
∈
V
∪
E
. For a capacity5