a cutting plane algorithm for the capacitated connected

Noname manuscript No.(will be inserted by the editor)

A cutting plane algorithm for the capacitatedconnected facility location problem

Stefan Gollowitzer · Bernard Gendron ·Ivana Ljubic

Received: date / Accepted: date

Abstract We consider a network design problem that arises in the cost-optimal design of last mile telecommunication networks. It extends the Con-nected Facility Location problem by introducing capacities on the facilities andlinks of the networks. It combines aspects of the capacitated network designproblem and the single-source capacitated facility location problem We referto it as the Capacitated Connected Facility Location Problem. We develop abasic integer programming model based on single-commodity flows. Based onvalid inequalities for the capacitated network design problem and the single-source capacitated facility location problem we derive several (new) classesof valid inequalities for the Capacitated Connected Facility Location Prob-lem including cut set inequalities, cover inequalities and combinations thereof.We use them in a branch-and-cut framework and show their applicability andefficacy on a set of real-world instances.

Keywords Capacitated Network Design · Facility Location · ConnectedFacility Location · Mixed Integer Programming Models · Telecommunications

S. Gollowitzer and I. LjubicDepartment of Statistics and Operations Research, Faculty of Business, Economics, andStatistics, University of Vienna, Brunner Straße 72, 1210 Vienna, AustriaTel.: +43-1-4277-38663Fax: +43-1-4277-38699E-mail: {stefan.gollowitzer,ivana.ljubic}@univie.ac.at

B. GendronInteruniversity Research Centre on Enterprise Networks, Logistics and Transportation(CIRRELT), and Department of Computer Science and Operations Research, Universitede Montreal, C.P. 6128, Succursale Centre-Ville, Montreal, Canada H3C 3J7E-mail: [email protected]

2 Stefan Gollowitzer et al.

1 Introduction

Given a set of customers, a set of potential facility locations and some inter-connection nodes, the goal of the Connected Facility Location problem (ConFL)is to find the minimum-cost way of assigning each customer to exactly oneopen facility, and connecting the open facilities via a Steiner tree. The sum ofcosts for the Steiner tree, the facility opening costs and the assignment costsneeds to be minimized. This problem models a network design problem thatarises in the design of last mile telecommunication networks when the fiberto the curb (FTTC) deployment strategy is applied (see, e.g., [19]). Contraryto the fiber to the home strategy, where each customer, i.e., household, hasits own fiber-optic uplink, in the FTTC strategy some of the existing copperwire infrastructure is used. More precisely, in an FTTC network, fiber op-tic cables run from a central office to a cabinet serving a neighborhood. Endusers connect to this cabinet using the existing copper connections. Expen-sive switching devices are installed in these cabinets. The usage of the last dmeters of copper wire between the customer and a switching device may sig-nificantly reduce deployment costs while still enabling broadband connectionsof reasonable quality.

In more detailed planning of FTTC networks, capacities of the links andof multiplexer devices are limited and this aspect was not captured by theConFL variants studied in the literature so far. In this paper we consider anew capacitated variant of the ConFL problem, that we will refer to as theCapacitated Connected Facility Location Problem (CapConFL).

In a typical application from telecommunications (see, e.g., Wassermann[35] for more details regarding input parameters), demands of customers aregiven as the number of twisted copper lines that are to be “served” at therespective customer location. Switching (or multiplexer) devices have bothcapacity and demand. Capacity is defined in terms of the number of twistedcopper lines a device can serve. The demand of a switching device is defined asthe number of fiber-optic uplinks required to connect the device to the centraloffice (which is further connected to the backbone network). The number ofuplinks is fixed for each device and independent of the number of customersthat are finally assigned to it. The CapConFL consists of deciding on thelocation of switching devices, the assignment of customers to these devicesand the routing of the uplinks from the switching devices to the central office,while minimizing the overall investment costs.

1.1 Problem definition

More formally, CapConFL can be defined as follows. The input is a graphG = (V,ES∪AR) with the set of nodes V partitioned into the set of customers(R), the set of potential facility locations (F ) and the set of potential Steinernodes (V \ (F ∪R)). A root node r ∈ V \ (F ∪R) represents the connection toa higher order (e.g., backbone) network. The network GS = (VS , ES), where

A cutting plane algorithm for the capacitated connected facility location problem 3

VS := V \R and ES := {e = {i, j} ∈ E | i, j ∈ VS} is called the core network.The assignment network GR = (F ∪ R,AR) consists of directed arcs betweenpotential facilities and customers, i.e., AR = {(i, k) | i ∈ F, k ∈ R}. Thefollowing input parameters are associated to the network:

– Facility opening cost fi ≥ 0, capacity vi > 0 and demand di > 0 for eachi ∈ F .

– Arc cost ce ≥ 0 and capacity ue > 0 for each e ∈ ES .– Assignment cost cij ≥ 0 for each (i, j) ∈ AR.– Customer demand bk > 0 for each k ∈ R.

The goal is to find a subnetwork of G consisting of the set of open facilitiesF ′, the set of core edges E′S and the set of assignment arcs A′R such that:

(P1) Each customer is assigned to exactly one open facility using arcs fromA′R.

(P2) The sum of customers’ demands assigned to a facility i does not exceedits capacity vi.

(P3) In the core subnetwork induced by E′S , we can simultaneously route theflow from the root node to satisfy the demand of all open facilities, withoutviolating the edge capacities.

(P4) The sum of assignment, facility opening and edge costs, given by∑e∈E′S

ce+∑i∈F ′ fi +

∑(i,j)∈A′R

cij , is minimized.

Obviously, by setting capacities ue = ∞, for all e ∈ ES and vi = ∞,for all i ∈ F , we obtain the previously studied ConFL problem. Figure 1illustrates solutions for ConFL and CapConFL. Squares and triangles denotefacilities and customers, respectively. A black fill indicates that a facility isopen. A diamond denotes the root node. Solid edges are in the core network,dotted edges represent the assignments. In the CapConFL the limited facilitycapacities require two additional open facilities and a different assignment ofcustomers to facilities. The limited edge capacities require additional edges inthe core network.

Notice that ConFL combines the Steiner tree problem and the uncapaci-tated facility location problem. On the other hand, CapConFL combines thesingle-source capacitated network design problem with the single-source ca-pacitated facility location problem. To see this, consider a feasible CapConFLinstance whose core graph has a star topology. One can easily transform thisinput graph into an instance of the single-source capacitated facility locationproblem: the facility opening costs for each i ∈ F are now defined as ce + fiwhere ce is the corresponding adjacent edge, and the assignment graph remainsunchanged. Similarly, a feasible CapConFL instance in which the assignmentarcs are such that each customer is adjacent to exactly one facility can bereduced into an instance of the single-source capacitated network design prob-lem.


(a) ConFL solution (b) CapConFL solution

Fig. 1 Feasible solutions of the ConFL and CapConFL problem, respectively.

1.2 Literature review

Since CapConFL has not been considered before, we provide a detailed liter-ature overview of three closely related problems: connected facility location,capacitated network design and single-source capacitated facility location.

1.2.1 Connected Facility Location

Early work on ConFL mainly includes approximation algorithms. ConFL canbe approximated within a constant ratio and the currently best-known ap-proximation ratio is provided by Eisenbrand et al. [14]. Recently, heuristicapproaches have been proposed by Ljubic [28] and Bardossy and Raghavan[3]. Gollowitzer and Ljubic [19] present and compare several formulations forConFL, both theoretically and computationally. Some of these results will bediscussed and related to the CapConFL later on. Arulselvan et al. [2] consider atime-dependent variant of the ConFL and present a branch-and-cut approachbased on cover, cut set cover and degree balance inequalities. Leitner and Raidl[27] propose a branch-and-cut-and-price approach for a variant of ConFL withcapacities on facilities. Cutting planes are used to ensure paths between theroot and open facilities, while column generation is used for selecting openfacilities and assigning customers to them.

1.2.2 (Single-Source) Capacitated Network Design Problems (CNDP)

In a typical CNDP setting, a network is given with a limited capacity availableon each edge. A subset of edges of minimum cost needs to be installed in thenetwork such that commodities with multiple origins and multiple destinationscan be routed through the network without violating installed edge capacities.There exists a large body of work on the CNDP and related problems.


It includes exact methods based on Lagrangian relaxation or decomposition[16, 22, 34, 8, 26, 15], heuristic methods based on tabu search, neighborhoodsearch, slope scaling, local branching and Lagrangian relaxation [10, 9, 17, 18,11, 25, 32]. Recent developments comprise a theoretical study and comparisonof Benders, metric and cut set inequalities [7] and a hybrid method com-bining mathematical programming and neighbourhood search techniques [20].Finally, Chouman et al. [5] present a branch-and-cut approach that comparesseveral families of valid inequalities for the CNDP.

A generalization of the single-source CNDP is the Local Access NetworkDesign problem (LAN). In this problem multiple copies of each edge are avail-able. The Local Access Network Design problem was studied by Raghavan andStanojevic [31], Salman et al. [33] and Ljubic et al. [29].

1.2.3 The Single-Source Capacitated Facility Location Problem (SSCFLP)

Aardal et al. [1] and Deng and Simchi-Levi [12] proposed MIP models and stud-ied the corresponding polyhedra of the SSCFLP and related problems. Holm-berg et al. [21] present a branch-and-bound method based on a Lagrangeanheuristic, Diaz and Fernandez [13] develop a branch-and-price approach basedon a decomposition of the SSCFLP and Contreras and Dıaz [6] propose ascatter search heuristic. Ceselli et al. [4] give an exhaustive computationalevaluation of branch-and-cut and branch-and-price approaches for a generalclass of facility location problems that includes the SSCFLP.

1.3 Contribution and outline

In Section 2 we introduce a basic integer programming model for CapConFLand discuss the relation of CapConFL and the Connected Facility Locationproblem. In particular, we show that a domination result between two setsof valid inequalities for ConFL does not hold for CapConFL. In Section 3 wederive cover and extended cover inequalities for the various knapsack type con-straints in our model. In addition, we provide two generalizations of recentlyproposed cut set cover inequalities and cover inequalities for single cut sets.Separation procedures for these valid inequalities are discussed in Section 4.In Section 5 we illustrate the effectiveness of the proposed model and the validinequalities by computational experiments on a set of new, realistic benchmarkinstances based on real data. Conclusions are given in Section 6.

2 Mixed integer programming models

In this section we introduce a first basic model for the CapConFL. It is basedon models familiar in the context of the SSCFLP and the CNDP. We thenstrengthen this model using concepts known from the Connected Facility Lo-cation problem [19].


Since all demands of open facilities have to be routed from a single sourcenode, it can be shown (see, e.g., [29]) that without loss of generality we canreplace the undirected core network GS by a bidirected graph in which eachedge e ∈ ES is replaced by two directed arcs, except for the edges adjacentto the root node, where it is sufficient to consider outgoing arcs from r. Theset of arcs of the bidirected core network will be denoted by AS . Since theflow routed through an edge will always be routed in one of the two oppositedirections, we define cost and capacities as cij = ce and uij = ue, respectively,for each e = {i, j} in ES . The union of core and assignment arcs is denotedby A = AS ∪AR. For a set of customers J ⊂ R we denote the set of facilitiesthat can serve these customers by F (J) =

⋃k∈J F (k), where F (k) := {i ∈

F : (i, k) ∈ AR}. Likewise, for I ⊂ F we denote by R(I) =⋃i∈I R(i) where

R(i) := {k ∈ R : (i, k) ∈ AR}. For W ⊂ V we denote the set of ingoing arcsby δ−(W ).

2.1 The basic MIP model

In our models we will use the following binary decision variables:

xij =

{1, if arc (i, j) is installed0, else

∀(i, j) ∈ A

zi =

{1, if facility i is installed0, else

∀i ∈ F

In addition, continuous flow variables gij indicate the total amount of flowbetween the root r and all open facilities in F routed through arc (i, j) ∈ A.

The following model combines the single-commodity flow (SCF) formula-tion for the CNDP (see, e.g., [31, 33]) with a formulation for the SSCFLP (see,e.g., [21]):

(SCF ) min∑ij∈A

cijxij+∑i∈F

fizi

s.t.∑ji∈AS

gji −∑ij∈AS

gij =

dlzl−

∑l∈F dlzl

0

i = li = relse

∀i ∈ VS (1a)

0 ≤ gij ≤ uijxij ∀(i, j) ∈ AS (1b)∑k∈R(i)

bkxik ≤ vizi ∀i ∈ F (1c)

xik ≤ zi ∀i ∈ F,∀k ∈ R(i) (1d)∑i∈F (k)

xik = 1 ∀k ∈ R (1e)

xij ∈ {0, 1} ∀(i, j) ∈ A (1f)


zi ∈ {0, 1} ∀i ∈ F (1g)

Constraints (1c)-(1e) are the strong relaxation of the SSCFLP. The assign-ment constraints (1e) model property (P1) and constraints (1c)-(1d) ensureproperty (P2). In constraints (1a)-(1b) we use the single-commodity flow vari-ables to ensure property (P3). This model is intuitive, but it provides weaklower bounds, due to the following facts: 1) big-M constraints (1b) are usedto model the arc capacities, and 2) the connectivity between the root and theopen facilities, rather than between the root and the customers, is required.The model is impractical to solve in a branch-and-bound framework, even formedium sized instances (see, e.g., Putz [30]).

By using the following capacitated cut set inequalities to replace constraints(1a) and (1b), we can project out the flow variables from the previous model(see, e.g., Ljubic et al. [29]):∑

ij∈δ−(W )

uijxij ≥∑

l∈F∩W

dlzl ∀W ⊆ VS \ {r} (CutSCF )

The obtained model contains an exponential number of inequalities and pro-vides the same lower bounds as the corresponding flow model. However, in-equalities (CutSCF ) can be strengthened as follows:∑

ij∈δ−(W )

min(uij ,∑

l∈F∩W

dl)xij ≥∑

l∈F∩W

dlzl ∀W ⊆ VS \ {r}

2.2 Relations to Connected Facility Location and cut set inequalities

Gollowitzer and Ljubic [19] studied MIP formulations for ConFL and provideda complete hierarchy of several MIP formulations with respect to the qualityof their LP-bounds. Among others, two cut set-based formulations for ConFLwere described. The models differ in the way they require connectivity.

In the first model, connectivity is ensured between the root and any openfacility as follows:∑

ij∈δ−(W )

xij ≥ zl ∀W ⊆ VS \ {r}, ∀l ∈W ∩ F (CutZ)

These inequalities state that for each open facility the edges on at least onepath between the root node and the respective facility need to be installed.Additional assignment constraints (1d) and (1e) are required between the fa-cilities and the customers.

The second model replaces constraints (CutZ) by the following cut setinequalities that ensure connectivity between the root and every customer:∑

ij∈δ−(W )

xij ≥ 1 ∀W ⊆ V \ {r},W ∩R 6= ∅ (CutX)


For ConFL it was shown that the second model provides theoretically strongerlower bounds, but is computationally outperformed by the first model on theset of benchmark instances considered there.

Both sets of inequalities, (CutZ) and (CutX) are also valid for CapConFL.It is interesting to mention that, unlike for ConFL, for which inequalities (CutZ)are implied by the model with (CutX) constraints, the two families of inequal-ities can be used complementary to each other for CapConFL:

Lemma 1 Inequalities (CutZ) and (CutX) both strengthen the LP-relaxationof the basic model (SCF). However, the MIP models (SCF)+(CutX) and(SCF)+(CutZ) are incomparable w.r.t. the quality of their LP-bounds.

Proof It is not difficult to see that inequalities (CutZ) and (CutX) bothstrengthen the LP-relaxation of (SCF). To see that (CutZ) inequalities are notimplied by (SCF)+(CutX), consider the example shown in Figure 2. A vector(x, z) that satisfies (CutSCF ) is x12 = 0.75, x23 = x24 = 0.25, z3 = z4 = 0.75,x35 = x46 = 0.75 and x45 = x36 = 0.25. This solution is cut off by the (CutX)constraints x23 + x24 ≥ 1 and x12 ≥ 1. Finally, inequalities (CutZ) are notredundant for (SCF)+(CutX) since they ensure x23 + x24 ≥ 1.5 which furtherstrengthens the model.Conversely, the model (SCF)+(CutZ) does not imply (CutX) constraints,which follows from the previous results for ConFL in [19], i.e., a CapConFLinstance with sufficiently large capacities on arcs and facilities will have thedesired property.

1 2

3

4

5

6

b5 = 6

b6 = 6

v3 = 8

v4 = 8

d3 = 2

d4 = 2

u = 4

u = 6

u = 6

Fig. 2 Example for comparison of cut set inequalities

3 Valid Inequalities

For the well-known subproblems of CapConFL, SSCFLP and CNDP, severalsets of strengthening valid inequalities are known. We will review ideas thatseem relevant in the context of the CapConFL and propose several sets of newvalid inequalities based on the combination of the facility location and networkdesign aspect.


3.1 Cover inequalities for single facilities

Deng and Simchi-Levi [12] proposed cover inequalities for the SSCFLP withuniform capacities. These inequalities are better known in the context of gen-eral mixed integer programming to strengthen knapsack-type constraints. Wewill use the concept of extended cover inequalities (see, e.g., the recent workof Kaparis and Letchford [24]).

Consider an arbitrary potential facility node i ∈ F . We call a set R′ ⊆ R(i)a cover for i ∈ F if

∑k∈R′ bk > vi and minimal if

∑k∈R′ bk − b` ≤ vi for all

` ∈ R′. For a minimal cover R′, we define E(R′) = {k ∈ R(i) \ R′ : bk ≥ b∗},where b∗ = maxk∈R′ bk.

Let the set of all minimal covers of i ∈ F be denoted by MC(i). Then thefollowing extended knapsack cover inequalities are valid for the CapConFL:∑

j∈R′∪E(R′)

xij ≤ (|R′| − 1)zi ∀R′ ∈MC(i),∀i ∈ F (EKS)

3.2 Inequalities involving multiple facilities

We derive two new families of inequalities that are implied by the limitedcapacities of facilities and the limited number of assignments edges in AR.

3.2.1 Minimum cardinality inequalities on facilities

For a given set of customers J ⊂ R and the corresponding subset of facilitiesF (J), let p(J) be the minimum number of facilities in F (J) that is requiredto assign the customers in J in a feasible way, i.e., by respecting the allowedpossible assignments and satisfying the capacity constraints on the facilitiesin F (J). In other words, p(J) is the optimum solution of a capacitated bin-packing problem with the set of bins F (J), capacities vi for i ∈ F (J), the setof items J , demands bj for j ∈ J and such that each item j ∈ J is only allowedto be assigned to bins in F (j). W.l.o.g. we can assume that bk ≤ vi for all(i, k) ∈ AR and thus p({k}) = 1 for all k ∈ R and p(J) ≤ min{|F (J)|, |J |} forall J ⊆ R.

Then the following minimum cardinality inequalities are valid for the Cap-ConFL: ∑

i∈F (J)

zi ≥ p(J) ∀J ⊆ R (MCF )

3.2.2 (Extended) Cover inequalities on facilities

Next we apply the idea of cover inequalities to the relation of facility capacitiesand customer demands. Let again be J ⊆ R. We call a set F ′ ⊂ F (J) a capacitycover with respect to J if

∑i∈F (J)\F ′ vi < b(J) and we call it minimal if

vk +∑i∈F (J)\F ′ vi ≥ b(J) for all k ∈ F ′. Let CC(F (J)) denote the set of all


such capacity covers of F (J). We call the following set of constraints coverinequalities on facilities:∑

i∈F ′zi ≥ 1 ∀F ′ ∈ CC(F (J)),∀J ⊆ R (2)

Similar to the cover inequalities for single facilities we can extend the coversand obtain stronger inequalities. Let v∗ = maxi∈F ′ vi and let E(F ′) = {i ∈F (J) \ F ′ : vi ≥ v∗} be the set of remaining facilities from F (J) with acapacity of at least v∗. We refer to the following inequalities as extended coverinequalities on facilities:∑

i∈F ′∪E(F ′)

zi ≥ 1 + |E(F ′)| ∀F ′ ∈ CC(F (J)),∀J ⊆ R (CovF )

To see that these inequalities are valid we can rewrite inequalities (2) as∑i∈F ′(1 − zi) ≤ |F ′| − 1. The corresponding extended cover inequality is

then ∑i∈F ′∪E(F ′)

(1− zi) ≤ |F ′| − 1.

Rewriting this inequality gives (CovF ).

The sets of inequalities (MCF ) and (CovF ) do not contain each other as thefollowing counterexamples show. In the example in Figure 3(a) a valid (CovF )inequality is z1+z2 ≥ 2, while the (MCF ) inequalities only ensure z1+z2+z3 ≥2. On the contrary, for the example given in Figure 3(b) the (CovF ) inequalitiesare z1 +z2 ≥ 1 and z1 +z3 ≥ 1, but they are strictly dominated by the (MCF )inequality z1 + z2 + z3 ≥ 2 that also implies z2 + z3 ≥ 1.

1

2

3

4

5

6

b4 = 15

b5 = 12

b6 = 12

v1 = 25

v2 = 25

v3 = 12

(a) Example 1

1

2

3

4

5

6

b4 = 16

b5 = 16

b6 = 17

v1 = 64

v2 = 32

v3 = 32

(b) Example 2

Fig. 3 Counterexamples for comparison of (MCF ) and (CovF )


3.2.3 General representation of cover inequalities on facilities

Consider now a general valid inequality of type∑i∈F

zi ≥ p (3)

defined for a set F ⊆ F and p ≥ 1. For p = 1 we have the simple coverinequalities (2) (i.e., F ∈ CC(F (J))) and, for p ≥ 2, inequalities of type (MCF )and (CovF ) belong to this family, i.e., we have F ∈ F (J)∪{F ′ ∪E(F ′) | F ′ ∈CC(F (J))}, for J ⊆ R. The following family of general cover inequalities onfacilities is then also valid for our problem:∑

i∈F

zi ≥ 1 ∀F ⊆ F, |F ∩ F | ≥ |F | − p+ 1 (Covgen)

It is not difficult to see that the latter inequalities are implied by (3). However,they are of particular interest when combined with cut set inequalities, asexplained below.

3.3 Cut set cover inequalities

This new family of valid inequalities combines cut set inequalities with the gen-eral cover inequalities for facilities of the form (Covgen). Inequalities (Covgen)state that at least one facility in F needs to be opened in a feasible solution.Consequently, for every subset of nodes W ⊂ V containing all nodes in F ,at least one ingoing arc needs to be installed. Let F denote the family of allsubsets of facilities for which (Covgen) is valid:

F =⋃J⊆R

F (J) ∪ {F ′ ∪ E(F ′) | F ′ ∈ CC(F (J))}

and let

p(F ) =

{1 + |E(F ′)|, F = F ′ ∪ E(F ′), F ′ ∈ CC(F (J))p(J), F = F (J)

for all F ∈ F . The following cut set cover inequalities are valid for CapConFLand not implied by any of the previously described sets of constraints:∑ij∈δ−(W )

xij ≥ 1 ∀F ⊆W∩F, |F ∩ F |≥|F | − p(F ) + 1, F ∈ F (CutCov)

Inequalities (CutCov) are a generalization of the previously introduced cutset cover inequalities for the incremental ConFL studied in Arulselvan et al.[2]. Figure 4 illustrates inequalities (CutCov) for two different subsets W anda cover inequality z1 + z2 ≥ 1 of type (CovF ). Figure 5 illustrates inequali-ties (CutCov) for the minimum cardinality inequality z1 + z2 + z3 ≥ 2.


r

1

2

3

4

5

6

W1W2

Fig. 4 Example for cut set cover inequalities (CutCov) derived from an inequality (CovF ).

r

1

2

3

4

5

6

W1

W2

Fig. 5 Example for cut set cover inequalities (CutCov) derived from an inequality (MCF ).

3.4 Cover inequalities for single cut sets

The following set of valid inequalities generalizes the cover inequalities knownfor the capacitated network design problem studied in Chouman et al. [5].Consider a (CutSCF ) cut set inequality

∑ij∈δ−(W ) uijxij ≥

∑l∈F∩W dlzl de-

fined by a cut set δ−(W ) for W ⊆ V \ {r}. Let F ′ ⊆ F ∩W and d(F ′) =∑l∈F ′ dl. A set C ⊂ δ−(W ) is called a cover with respect to δ−(W ) and F ′,

if∑ij∈δ−(W )\C uij < d(F ′) and a minimal cover if, in addition,∑

ij∈δ−(W )\C

uij + ulk ≥ d(F ′) ∀(l, k) ∈ C.

Let MC(W,F ′) denote the set of all minimal covers with respect to δ−(W )and F ′. Then the following cover inequalities on single cut sets are valid forthe CapConFL:∑

ij∈Cxij ≥ 1 +

∑l∈F ′

(zl − 1) ∀C ∈MC(W,F ′). (Covδ−(W ))

Figure 6 illustrates inequalities (Covδ−(W )). Edge (b, d) is a cover withrespect to W = {1, 2, 3, d, e, f} and F ′ = {2, 3}.

4 Separation procedures

In this section we describe the separation procedures used in our branch-and-cut algorithm. We refer to the variable values of the current fractional solutionby (x, z).


r

1

2

3

e

c

b

fd

d1 = 5

d2 = 7

d3 = 12u = 7

u = 9

u = 30

Fig. 6 Illustration of cut set cover inequalities

4.1 Separation of inequalities (CutSCF ) and (CutZ)

Inequalities (CutSCF ) can be separated in polynomial time (see also Ljubicet al. [29]). We define the support graph G′ = (V ′, A′) where V ′ := VS ∪ twith an additional sink node t, A′ := AS ∪At and At := {(i, t) | i ∈ F, zi > 0}.We define capacities on arcs as uij xij for each arc ij ∈ AS and dizi for eacharc it ∈ At. We calculate the minimum cut between r and t in G′. Let δ−(W )denote the arcs of this cut. If δ−(W ) ∩ AS 6= ∅ and

∑ij∈δ−(W )∩AS

uij xij <∑i∈W∩F dizi we have detected a violated inequality (CutSCF ).Inequalities (CutZ) can be separated in similar fashion (see also Gollowitzer

and Ljubic [19]). The support graph in this case is the bidirected core network(VS , AS) with arc capacities set to xij for each arc ij ∈ AS . A minimum cut inAS between r and l ∈ F with a weight of less than zl corresponds to a violatedinequality (CutZ).

4.2 Separation of inequalities (CutX)

For the separation of (CutX) inequalities we define a support graph Gj foreach j ∈ R. Thereby, Gj = (V ∪ {j},AS ∪Aj) where Aj = {(i, j) | i ∈ F (j)}.Capacities on the arcs from AS ∪ Aj are set to xij . Each minimum cut in Gjbetween r and j ∈ R whose weight is less than 1 corresponds to a violatedinequality (CutX).

If the number of customers is large, complete separation of inequalities (CutX)is very time-consuming. We therefore reduce the set of customers consideredin the separation to a subset that still ensures that all violated inequalitiesare identified. A customer c1 ∈ C is ignored if there exists another customerc2 ∈ C such that F (c2) ⊂ F (c1). If sets F (ci) are identical for all ci ∈ C ⊆ Conly one customer in C is considered.

4.3 Separation of inequalities (EKS)

For a fractional point (x, z) and for each i ∈ F = {i ∈ F | zi > 0}, theseparation of (simple, non-extended) inequalities (EKS) is equivalent to solving


a knapsack problem which is described by the following integer program:

min α =∑j∈R

(zi − xij)sj

s.t.∑j∈R

bjsj > vi

sj ∈ {0, 1} ∀j ∈ R

If α < zi a simple, non-extended cover inequality is violated.Instead of solving the separation problem exactly (e.g., using a dynamic

programming procedure), we use a heuristic separation proposed by Kaparisand Letchford [24] that was shown to be effective in directly detecting extendedknapsack cover inequalities. Adapted to our (EKS) inequalities, this procedureconsists of the following steps, executed for each i ∈ F :

1. Sort the items in R(i) in non-decreasing order of (zi − xij)/bj , and storethem in a list L. Initialize the cover R′ as the empty set and initializeb∗ = vi.

2. Remove an item from the head of the sorted list L. If its weight is largerthan b∗, ignore it, otherwise insert it into R′. If R′ is now a cover, go tostep 4.

3. If L is empty, stop. Otherwise, return to step 2.4. If the extended cover inequality corresponding to R′ is violated by (x, z),

output it.5. Let k∗ = arg maxj∈R′ bj be the customer in R′ with the highest demand.

Set b∗ = bk∗ and delete k∗ from R′. Return to step 2.

In fact, we perform two variants of this algorithm. The one stated above andone where the items in R(i) are sorted in non-increasing order of xij .

4.4 Separation of inequalities (MCF ) and (CovF )

We consider subsets of facilities F ′ ∈ FC := F1 ∪ F2, where F1 := {F (k) |k ∈ R} and F2 := {F (k1) ∪ F (k2) | |F (k1) ∩ F (k2)|/min(|F (k1)|, |F (k2)|) ≥0.5, k1, k2 ∈ R}, i.e., F2 contains unions of F (k1) and F (k2) such that at leasthalf the facilities of either F (k1) or F (k2) are common to both these sets. Foreach F ′ we define the subset of customers to be considered in the separationof (MCF ) and (CovF ) inequalities as J(F ′) := {k′ ∈ R | F (k′) ⊆ F ′}.

4.4.1 Separation of inequalities (MCF )

We calculate p(J) for J and F (J) by solving the following bin-packing problemwith assignment restrictions and non-uniform bin capacities:

p(J) = min∑

i∈F (J)

ti


s.t.∑k∈R(i)

bksik ≤ viti ∀i ∈ F (J)

sik ≤ ti ∀k ∈ J, ∀i ∈ F (k)∑i∈F (k)

sik = 1 ∀k ∈ J

sik ∈ {0, 1} ∀k ∈ J, ∀i ∈ F (k)ti ∈ {0, 1} ∀i ∈ F (J)

We consider F ′ ∈ FC as candidate sets for F (J) and determine J = J(F ′)as described in the previous paragraph. The values of p(J) are calculatedfor all such J by means of a general purpose solver during preprocessing. Inthe separation procedure we repeatedly check whether the current fractionalsolution violates any of the stored inequalities (MCF ). By doing so we considerat most |FC | ≤ |R|+ |R|2 inequalities of type (MCF ).

4.4.2 Separation of inequalities (CovF )

Given J ⊆ R and F (J), the separation of (simple, non-extended) covers onfacilities (2) is equivalent to solving the following knapsack problem:

min α =∑

i∈F (J)

zisi

s.t.∑

i∈F (J)

visi >∑

i∈F (J)

vi − b(J)

si ∈ {0, 1} ∀i ∈ F (J)

If α < 1, an inequality (2) is violated.We consider F (J) for J = J(F ′) and F ′ ∈ FC . To find covers in CC(F (J))

we use the separation procedure described in Section 4.3 with the followingmodifications: The facilities in F (J) are ordered according to zi/vi in non-decreasing fashion and b∗ is initialized with the maximum capacity of thefacilities in F (J).

4.5 Separation of inequalities (CutCov)

In the separation of (CutCov) we consider all inequalities of the form (3)that were found by the separation procedure for (CovF ) and (MCF ). For thecorresponding set of facilities F and right-hand side p we randomly generate upto p sets F ⊆ F such that |F | = |F |−p+1. We separate inequalities (CutCov)by running a maximum flow algorithm on graph G′ defined as in Section 4.1,but with capacities of 1 on arcs (i, t) if i ∈ F and 0 if i 6∈ F .


4.6 Separation of inequalities (Covδ−(W ))

Given a cut set W ⊆ V \ {r} and the set of facilities contained in that cutset, F ′ = W ∩F , a violated cut set cover inequality is detected by solving thefollowing integer program:

min α =∑

ij∈δ(W )

xijsij +∑l∈F ′

(1− zl)tl

s.t.∑

ij∈δ−(W )

uijsij +∑l∈F ′

dltl >∑

ij∈δ−(W )

uij

sij ∈ {0, 1} ∀(i, j) ∈ δ−(W )tl ∈ {0, 1} ∀l ∈ F ′

A (Covδ−(W )) inequality is violated if α < 1.We separate inequalities (Covδ−(W )) as follows: All cut sets W that are

obtained during the separation of inequalities (CutSCF ) are kept in a pool.We choose F ′ = {l ∈ F ∩W | zl > 0.1}. Then we use the following heuristicprocedure to find minimal covers C ∈MC(W,F ′′), where F ′′ ⊆ F ′:

1. Sort the items in δ−(W ) and F ′ in non-decreasing order of (1− zi)/di andxij/uij and store them in a list L. Initialize the cover C and F ′′ as emptysets and initialize u∗ =

∑ij∈δ−(W ) uij .

2. Remove an item from the head of the sorted list L.(a) If it is an arc and its capacity is larger than u∗, ignore it, otherwise

insert it into C.(b) If it is a facility insert it into F ′′.If C is now a cover with respect to δ−(W ) and F ′′, go to step 4.

3. If L is empty, stop. Otherwise, return to step 2.4. If the cover inequality corresponding to C ∈ MC(W,F ′′) is violated by

(x, z), output it.5. Let (i∗, j∗) = arg maxij∈C uij be the arc in C with the highest capacity.

Set u∗ = ui∗j∗ and delete (i∗, j∗) from C. Return to step 2.

5 Computational results

In this section we report the results of our computational experiments. Theywere performed on a desktop machine with an 8-core Intel Core i7 CPU at 2.80GHz and 8 GB RAM. Each run was performed on a single core. We used theCPLEX [23] branch-and-cut framework, version 12.2. All cutting plane gener-ation procedures provided by CPLEX are turned off unless stated explicitly.All heuristics provided by CPLEX are turned off. The other parameters areset to their default values.


5.1 Branch-and-cut framework

The settings described in this section are the result of our preliminary testing.To reduce the number of constraints that need to be identified by our

separation routines we add degree balance constraints and subtour eliminationconstraints for cycles of size two to our model:

xji ≤ x(δ+(i)) + zi ∀(j, i) ∈ AS , i ∈ F (4a)

xji ≤ x(δ+(i)) ∀(j, i) ∈ AS , i ∈ VS \ (F ∪ {r}) (4b)

zi ≤ x(δ−(i)) ∀i ∈ F (4c)

xij ≤ x(δ−(i)) ∀(i, j) ∈ AS , i ∈ VS \ {r} (4d)xij + xji ≤ 1 ∀(i, j) ∈ AS , i < j, i, j 6= r (4e)

In order to reduce the size of the linear programs solved throughout theprocess we relax constraints (1d) and add them only if they are violated.Separation procedures are called in the following order: (EKS) - (CovF ) -(MCF ) - (1d) - (Covδ−(W )) - (CutCov) - (CutZ) - (CutX) - (CutSCF ). Toprevent a tailing off effect of the separation procedures we stop separatingvalid inequalities if the lower bound has improved by less than 0.05% for thelast 10 calls of the separation procedures. We apply this rule at each node ofthe branch-and-bound tree.

Inequalities (CovF ), (MCF ), (Covδ−(W )), (CutCov) and (CutSCF ) are onlyseparated at the root node of the branch-and-bound tree. Inequalities (EKS)and (1d) are separated at every node, separation of (CutX) is done at every10th node and separation of (CutZ) is done at every 100th node.

To improve the computational efficiency of the separation procedures forcut set inequalities, we search for nested minimum cardinality cuts. To do so,all capacities in the respective separation graph are increased by some ε > 0.Thus, every detected violated cut contains the least possible number of arcs.We resolve the linear program after adding at most 30 violated inequalities ofany class. Finally, we randomly choose the target nodes to search for violatedcuts.

5.2 Instances

We generated a set of realistic benchmark instances derived from real worlddata, which were provided by an Austrian telecommunications provider. Thekey figures for the instances we use are listed in Table 1.

The real world data contain most of the information needed for completeCapConFL instances: The sets of facilities, Steiner nodes and edges of the corenetwork; a set of customers with associated demands; a set of assignment arcsconnecting customers and facilities, including their distance and an estimate ofthe bandwidth provided by the respective assignment arc; lengths of core edgesand assignment arcs. These inputs define five graphs with different topologies


that will be denoted by A, B, C, D and E. To complete the instances with respectto the input required by CapConFL we applied the following steps:

– For each instance a minimum customer bandwidth is selected, assignmentarcs that provide less than this bandwidth are removed. We chose 20, 25and 30 MBit/s and denote this by 20, 25 and 30 in the instance label.

– At most 20 assignment arcs per customer are considered.– Customers without assignment arcs are removed and facilities without as-

signment arcs are replaced by Steiner nodes.– Steiner nodes with a degree of two and their adjacent edges are replaced

by a single edge.– A technology for each facility is randomly selected. For FTTB instances

we consider the following combinations of capacity, demand and cost:(32, 4, 4000), (64, 5, 6000), (128, 7, 8000). For FTTC instances we choosebetween (64, 4, 13000), (128, 4, 16000) and (192, 4, 20000).

– Edge capacities are uniformly randomly selected from [0.7µ, 1.3µ], whereµ is equal to the demand of the smallest set of facilities needed to feasiblyassign the customers, given the facility capacities chosen before [8].

5.3 Comparison against basic model and general purpose solver

In the first part of our computational study we assess the influence of the cut-ting plane generation procedures built into CPLEX compared to the influenceof the valid inequalities proposed in this work. To this end we ran our modelwith the following different settings: Basic is the cut set based model corre-sponding to SCF, i.e., the model consisting of constraints (1c)-(1g), (CutSCF )and (4a)-(4e). Basic+CPX is the basic model with all CPLEX cuts turned on.All VI is the basic model with all valid inequalities from Section 3 added. AllVI+CPX is the basic model with all valid inequalities and CPLEX cuts turnedon.

In Table 1 we compare the LP gaps (gLP ) and time to solve the LP relax-ation (tLP ) for these four models. We calculated the gaps as (UB − LB)/UB ,where UB is the best known integer solution found in all our tests and LB isthe solution value of the LP relaxation of the respective model (after dynamicaddition of cuts). In the last two lines we show the mean and median of thevalues in the respective column. The best LP gap of the four models is shownin bold.

From the results in Table 1 we conclude that the model without validinequalities provides a weak LP bound with an LP gap of 14.28% on averageover the considered instance set. The cutting planes provided by CPLEX canreduce the LP gaps of the basic model by almost one half to an average of7.23%. This average gap is still substantial compared to 1.31% obtained by themodel that is strengthened by the valid inequalities proposed in this paper.Using CPLEX cuts in addition only improves the average gap to 1.13%.

A cutting plane algorithm for the capacitated connected facility location problem 19In

stan

cep

rop

erti

esB

asi

cB

asi

c+C

PX

All

VI

All

VI+

CP

X|V

||E

||F

||R

||E

S|

|ER|

UB

g LP

t LP

g LP

t LP

g LP

t LP

g LP

t LP

A20-FTTC

2405

12827

1504

805

1636

11191

3315257

12.9

810

7.6

745

3.5

633

3.5

636

A25-FTTC

2283

11349

1371

794

1525

9824

3717919

9.5

411

4.1

518

3.3

636

3.1

143

A30-FTTC

1949

7689

1053

694

1289

6400

3356273

8.6

25

4.7

412

3.4

529

2.7

738

B20-FTTC

2300

9695

1624

510

1824

7871

4601666

15.8

717

4.3

762

0.1

039

0.1

039

B25-FTTC

2265

9141

1580

507

1792

7349

5051559

14.4

213

5.3

729

0.1

536

0.0

741

B30-FTTC

1848

5610

1152

450

1432

4178

6030938

9.1

88

4.0

616

0.1

017

0.0

218

C20-FTTC

4742

21388

3237

1160

3720

17668

8897504

21.0

6149

11.1

2409

0.4

5141

0.3

6152

C25-FTTC

4498

17477

2961

1134

3498

13979

9734526

16.2

955

9.9

1228

0.5

7134

0.5

3145

C30-FTTC

3269

9303

1748

840

2557

6746

9332930

11.1

447

4.7

375

0.8

235

0.7

058

D20-FTTC

4042

24923

2241

1463

2661

22262

7962207

13.1

850

8.8

5156

1.9

363

1.9

079

D25-FTTC

3925

21403

2106

1449

2557

18846

9490801

8.8

638

4.3

176

1.3

262

1.0

174

D30-FTTC

3407

11539

1588

1213

2274

9265

9971466

4.0

312

2.6

630

0.7

261

0.4

573

E20-FTTC

3426

18045

2143

1038

2492

15553

4858219

23.9

5101

15.8

1302

1.0

661

0.8

693

E25-FTTC

3290

12574

1965

1023

2369

10205

7419660

13.2

158

5.1

3114

0.5

451

0.3

474

E30-FTTC

2149

4429

750

497

1747

2682

6082850

6.9

37

2.5

49

0.0

021

0.0

021

A20-FTTB

2405

12827

1504

805

1636

11191

2506131

16.0

010

9.5

637

3.2

340

2.4

248

A25-FTTB

2267

11284

1370

778

1525

9759

2741939

12.6

48

7.2

733

3.2

638

2.8

147

A30-FTTB

1949

7689

1053

694

1289

6400

2406722

10.0

14

3.7

517

2.7

326

2.1

437

B20-FTTB

2300

9695

1624

510

1824

7871

3781915

19.9

913

7.8

366

0.2

640

0.2

149

B25-FTTB

2265

9141

1580

507

1792

7349

4146838

18.2

211

5.7

731

0.1

737

0.0

737

B30-FTTB

1848

5610

1152

450

1432

4178

4797642

12.1

210

3.7

518

0.0

619

0.0

520

C20-FTTB

4742

21388

3237

1160

3720

17668

7337282

26.2

0176

14.1

1397

0.6

2158

0.5

6171

C25-FTTB

4498

17477

2961

1134

3498

13979

7836463

20.7

566

10.9

3229

0.7

8120

0.6

9159

C30-FTTB

3269

9303

1748

840

2557

6746

7332424

14.1

844

6.8

563

0.8

956

0.7

961

D20-FTTB

4042

24923

2241

1463

2661

22262

7056336

13.9

446

9.3

0131

1.0

657

1.0

268

D25-FTTB

3925

21403

2106

1449

2557

18846

8315107

10.2

639

6.2

278

1.4

465

1.2

071

D30-FTTB

3407

11539

1588

1213

2274

9265

8210407

4.6

711

3.0

528

0.5

650

0.4

160

E20-FTTB

3426

18045

2143

1038

2492

15553

3956572

29.9

8188

19.1

8413

1.3

864

1.2

388

E25-FTTB

3290

12574

1965

1023

2369

10205

5885571

19.5

399

9.2

7170

3.0

765

2.8

960

E30-FTTB

2149

4429

750

497

1747

2682

4651295

10.5

35

4.6

69

1.5

524

1.4

823

mea

n14.2

844

7.2

3110

1.3

156

1.1

366

med

ian

13.1

915

6.0

062

0.8

645

0.7

559

Table

1C

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pari

son

of

LP

rela

xati

on

low

erb

ou

nd

sw

ith

an

dw

ith

ou

tC

PL

EX

cuts


5.4 Influence of different sets of valid inequalities

In the second part of our computational study we assess the influence of thedifferent sets of valid inequalities proposed in this work. We compare five dif-ferent settings that differ by the sets of valid inequalities considered. For eachsetting we add a subset of valid inequalities to the basic model described above.Settings (CutZ), (CutX) and (CutZ)+(CutX) are self-explaining. Setting AllVI is defined as above and setting Most VI uses inequalities (CutZ), (CutX),(EKS), (CovF ) and (MCF ).

For each of these settings, Table 2 shows the gap of the linear programmingrelaxation, gLP , after the dynamic addition of cuts and calculated as in Table 1,the time needed to solve linear programming relaxation, tLP , and the numberof cutting planes added, Cuts. In the last two lines we show the mean andmedian of the values in the respective column. The best LP gap of all modelsis shown in bold.

We would like to point out several interesting aspects. The LP gaps ofsetting (CutZ) are substantially larger than the ones of all other settings.Surprisingly the same does not hold for setting (CutX), which on averagegives even stronger LP bounds than setting (CutZ)+(CutX). We trace thedifference between the gaps of (CutX) and (CutZ)+(CutX) to the criteriawe used to prevent a tailing off effect during separation. A comparison ofthe running times shows that separating valid inequalities with a differentstructure improves the overall running time of the LP relaxation. Approaches(CutZ) and (CutX) need 110 and 90 seconds on average, respectively, whereasapproach (CutZ)+(CutX) only takes 46 seconds to compute approximately thesame lower bounds as (CutX). The separation routines in approaches Most VIand All VI require an additional 10 and 12 seconds on average. Thereby, theaverage LP gaps are improved from 1.44% ((CutZ)+(CutX)) to 1.31% (MostVI and All VI ). However, All VI does not improve upon Most VI significantly.

There is a notable difference in the numbers of valid inequalities that weredetected during the LP relaxations of the different settings. By far the highestnumber of inequalities is found by setting (CutZ), even though the obtainedLP bound is comparably weak. This is consistent with the long running timeof the LP relaxation of setting (CutZ). Rather surprising is the fact, thatsetting All VI obtains the same LP bound as setting Most VI for 28 out of30 settings but the number of valid inequalities found by All VI is smaller for22 and larger for only 2 instances.

Table 3 shows the respective gap of the five different settings after 3, 10, 30and 60 minutes. For these results we calculate the gaps as (UB t − LB t)/UB t

where UB t is the best integer solution found by the respective setting after tminutes and LB t is the lower bound after t minutes. For each instance andrunning time the smallest gap of all five settings is indicated in bold. If nointeger solution is available after t minutes we indicate this by a dash in therespective column. For each setting and time t the last three lines of the tableindicate the mean and median of gaps over the instance set and how often therespective approach gives the smallest gap of all settings.

A cutting plane algorithm for the capacitated connected facility location problem 21(C

ut Z

)(C

ut X

)(C

ut Z

)+(C

ut X

)M

ost

VI

All

VI

g LP

t LP

Cuts

g LP

t LP

Cuts

g LP

t LP

Cuts

g LP

t LP

Cuts

g LP

t LP

Cuts

A20-FTTC

8.5

632

5711

3.5

748

3879

3.5

827

4168

3.5

631

4776

3.5

633

4523

A25-FTTC

4.4

820

4623

3.5

094

2984

3.5

131

3101

3.3

832

3528

3.3

636

3446

A30-FTTC

7.1

217

3565

4.0

547

2987

4.0

524

3144

3.4

527

3855

3.4

529

3660

B20-FTTC

11.8

5105

7323

0.1

539

4459

0.1

526

5100

0.1

039

5140

0.1

039

5140

B25-FTTC

10.4

026

5669

0.1

850

4442

0.1

833

4815

0.1

535

4908

0.1

536

4908

B30-FTTC

5.0

317

3447

0.1

224

2382

0.1

216

2661

0.1

017

2728

0.1

017

2728

C20-FTTC

12.9

8351

14191

0.5

4293

9042

0.5

4144

9813

0.4

5142

10114

0.4

5141

9831

C25-FTTC

8.3

2265

11612

0.6

2215

6947

0.6

2122

7298

0.5

6122

8107

0.5

7134

7710

C30-FTTC

4.9

162

4514

0.9

879

3290

0.9

832

3809

0.8

235

3873

0.8

235

3799

D20-FTTC

7.7

9232

13921

2.0

097

7561

2.0

051

8193

1.9

362

8751

1.9

363

8124

D25-FTTC

3.6

862

8784

1.3

368

5901

1.3

439

6056

1.3

262

6454

1.3

262

6529

D30-FTTC

1.2

850

6150

0.8

2109

4578

0.8

255

4737

0.7

258

5001

0.7

261

4974

E20-FTTC

13.6

9142

12891

1.1

0140

8633

1.1

058

9566

1.0

661

10021

1.0

661

10021

E25-FTTC

4.1

750

8515

0.5

762

6046

0.5

835

6303

0.5

450

6656

0.5

451

6656

E30-FTTC

2.2

912

2222

0.0

120

1682

0.0

114

1795

0.0

022

1960

0.0

021

1849

A20-FTTB

10.4

625

5426

3.4

667

3681

3.5

838

3640

3.2

340

4880

3.2

340

4394

A25-FTTB

5.0

920

3857

3.5

378

2740

3.5

729

2851

3.2

632

3789

3.2

638

3457

A30-FTTB

8.2

514

2755

3.5

926

2036

3.6

013

2133

2.7

322

3074

2.7

326

2832

B20-FTTB

15.8

274

7895

0.3

654

4497

0.3

634

4977

0.2

638

5365

0.2

640

4944

B25-FTTB

13.7

145

6044

0.2

650

4503

0.2

632

4948

0.1

737

5051

0.1

737

4913

B30-FTTB

7.2

016

3478

0.1

125

2387

0.1

118

2661

0.0

622

2601

0.0

619

2664

C20-FTTB

16.7

5657

15243

0.7

1298

8207

0.7

1148

8980

0.6

2156

9785

0.6

2158

9475

C25-FTTB

11.2

6306

12195

0.8

893

6765

0.8

867

7266

0.7

8118

7793

0.7

8120

7364

C30-FTTB

6.7

384

4785

1.1

093

3380

1.1

050

3693

0.9

057

3942

0.8

956

3756

D20-FTTB

8.3

9210

12375

1.1

065

5535

1.1

030

6459

1.0

659

7591

1.0

657

6476

D25-FTTB

4.5

675

8617

1.4

6154

4824

1.4

650

4950

1.4

459

6104

1.4

465

5565

D30-FTTB

1.3

428

6054

0.6

743

4030

0.6

724

4287

0.5

646

4543

0.5

650

4308

E20-FTTB

18.3

0213

12824

1.3

9139

7394

1.3

968

7869

1.3

874

8747

1.3

864

8379

E25-FTTB

8.4

182

8547

3.0

9113

5820

3.1

261

6010

3.0

750

6699

3.0

765

6443

E30-FTTB

5.3

620

2288

1.6

810

1617

1.6

813

1811

1.5

524

1937

1.5

524

1937

mea

n8.2

7110

7517

1.4

390

4740

1.4

446

5103

1.3

154

5592

1.3

156

5360

med

ian

8.0

256

6102

1.0

467

4478

1.0

433

4881

0.8

643

5026

0.8

645

4928

Table

2C

om

pari

son

of

LP

rela

xati

on

low

erb

ou

nd

sw

ith

diff

eren

tse

tsof

valid

ineq

ualiti

estu

rned

on

22 Stefan Gollowitzer et al.(C

ut Z

)(C

ut X

)(C

ut Z

)+(C

ut X

)M

ost

VI

All

VI

g 3g 1

0g 3

0g 6

0g 3

g 10

g 30

g 60

g 3g 1

0g 3

0g 6

0g 3

g 10

g 30

g 60

g 3g 1

0g 3

0g 6

0

A20-FTTC

7.2

85.4

44.9

74.1

2-

-4.7

64.7

64.2

54.2

24.0

33.4

54.3

44.2

24.2

14.1

83.6

83.6

63.6

23.6

1A25-FTTC

5.5

45.2

65.0

44.9

6-

-4.9

94.9

65.8

85.8

25.7

75.7

44.4

44.3

74.3

24.2

93.7

43.6

63.6

13.5

8A30-FTTC

6.6

55.8

25.3

15.1

7-

5.8

15.6

65.5

25.7

25.5

54.3

73.9

34.8

73.6

13.3

13.0

24.9

33.5

83.4

02.8

6B20-FTTC

11.5

95.5

53.4

42.1

30.0

70.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0B25-FTTC

7.5

84.6

52.2

51.2

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0B30-FTTC

1.4

80.6

90.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0C20-FTTC

-15.9

16.2

15.0

3-

-1.9

41.4

8-

1.8

31.7

90.6

1-

0.8

20.5

90.3

2-

1.0

00.5

20.4

6C25-FTTC

-5.8

54.0

93.7

2-

1.0

30.9

90.9

00.9

90.9

30.8

80.8

51.0

60.8

60.7

00.6

60.8

30.7

30.6

50.6

2C30-FTTC

5.2

23.3

73.0

02.6

81.4

71.4

21.3

31.2

41.3

41.1

91.1

01.0

10.9

50.8

20.3

00.0

50.9

10.7

30.1

30.0

0D20-FTTC

--

3.3

42.8

3-

-2.5

92.5

9-

2.4

92.4

62.4

2-

2.5

32.0

01.9

0-

2.1

52.1

02.0

7D25-FTTC

4.1

42.3

72.0

71.7

7-

-1.9

91.8

02.3

91.6

11.2

41.1

8-

1.6

21.1

81.0

6-

1.6

01.4

61.1

3D30-FTTC

1.5

31.1

90.9

60.8

9-

1.3

11.2

30.9

61.2

30.6

90.5

50.5

31.1

71.0

40.8

70.5

91.2

00.9

10.6

00.5

8E20-FTTC

-8.3

25.5

04.2

5-

1.9

81.7

71.6

61.7

91.4

31.2

21.1

61.7

31.2

10.8

50.8

01.7

31.2

10.8

50.8

0E25-FTTC

4.3

72.5

62.2

41.8

51.9

91.8

20.7

00.6

70.7

00.5

50.4

20.3

20.7

30.5

90.4

60.3

50.7

30.5

90.4

60.3

5E30-FTTC

0.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0A20-FTTB

8.3

17.2

95.5

65.2

1-

-5.0

75.0

55.7

35.6

85.6

25.4

9-

3.1

73.0

62.9

6-

2.7

02.5

32.4

3A25-FTTB

6.9

06.5

86.2

74.4

7-

3.5

23.4

53.3

74.8

24.7

34.5

94.4

63.5

03.2

63.0

12.9

73.3

83.1

93.0

53.0

2A30-FTTB

6.8

55.2

24.8

64.6

5-

6.1

35.9

85.9

55.0

64.8

84.7

84.7

32.9

42.6

62.2

92.1

02.8

42.6

32.4

22.3

4B20-FTTB

16.1

711.1

56.5

24.5

80.6

50.6

00.5

50.4

30.4

40.3

70.2

50.2

30.3

00.2

70.1

60.1

40.3

20.2

20.1

80.1

6B25-FTTB

10.9

97.8

75.2

24.1

10.5

80.4

70.3

80.3

40.2

30.1

00.0

50.0

30.0

40.0

00.0

00.0

00.0

60.0

20.0

00.0

0B30-FTTB

3.5

71.4

10.0

00.0

00.0

20.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0C20-FTTB

--

11.4

38.6

9-

-1.3

21.3

1-

1.1

30.8

80.8

5-

0.7

60.7

20.5

8-

0.8

00.7

80.7

2C25-FTTB

--

5.4

34.6

8-

-1.8

41.1

9-

1.3

21.0

21.0

0-

0.8

90.6

90.6

5-

0.8

50.6

10.4

6C30-FTTB

7.3

22.7

91.1

60.8

1-

0.6

90.6

60.6

20.5

90.4

30.3

70.3

40.4

20.3

50.2

70.2

50.3

30.2

50.2

10.1

6D20-FTTB

-4.6

82.7

32.2

5-

--

1.4

1-

1.2

11.1

91.1

7-

1.3

91.1

71.1

4-

1.1

91.0

71.0

4D25-FTTB

-2.9

62.1

01.8

5-

--

2.2

9-

1.7

71.7

21.7

0-

1.9

31.4

91.4

5-

2.2

81.3

11.2

9D30-FTTB

1.4

01.1

20.9

20.8

0-

0.8

30.7

00.6

20.8

70.5

80.4

30.4

10.6

40.5

10.3

60.3

30.5

40.4

70.4

00.3

3E20-FTTB

-10.1

86.0

94.4

7-

-1.6

61.6

32.3

31.5

21.3

61.3

52.5

71.7

91.3

41.2

21.6

11.5

41.3

31.3

1E25-FTTB

-3.9

22.3

92.1

3-

1.6

80.8

70.7

10.7

70.5

10.4

40.4

11.4

10.6

20.4

50.2

80.9

10.5

70.5

00.4

4E30-FTTB

2.1

81.1

50.1

20.1

00.4

00.2

00.1

60.1

50.1

40.1

20.1

00.0

80.1

40.0

40.0

00.0

00.1

40.0

40.0

00.0

0m

ean

--

3.6

42.9

8-

--

1.7

2-

1.6

91.5

51.4

5-

1.3

11.1

31.0

4-

1.2

21.0

60.9

9m

edia

n-

-3.3

92.7

5-

--

1.2

1-

1.1

60.9

50.8

5-

0.8

40.7

00.5

8-

0.8

30.6

10.5

2b

est

11

33

35

55

10

10

88

11

10

16

18

16

22

19

18

Table

3G

ap

sto

mod

elu

pp

erb

ou

nd


Contrary to what the LP gaps in Table 2 suggest the setting All VI withall valid inequalities enabled outperforms the other settings on a majority ofinstances. The performance of setting Most VI is only slightly worse (0.09%,0.07% and 0.05% larger gap after 10, 30 and 60 minutes). The other settingsperform significantly worse with between 0.46% and 2.58% larger gaps on av-erage.

In Figures 7 and 8 we give a graphical illustration of the numbers reportedin Table 3. The coordinates of each mark indicate how many out of 30 in-stances (ordinate axis) were solved within a given optimality gap (abscissa).Figure 7 shows the performance after 3 and 10 minutes and Figure 8 showsthe performance after 30 and 60 minutes.

6 Conclusions and future research

In this paper we introduced the Capacitated Connected Facility Location prob-lem. We described various sets of cut set, minimum cardinality, cover and cutset cover inequalities to strengthen a basic integer programming model. Af-ter a detailed discussion of separation procedures we reported the results ofour computational experiments. These confirmed that the proposed approachfinds solutions within a small optimality gap averaging to less than 2% for aset of realistic new benchmark instances.

The goal of this work was to contribute to a better understanding of thecomputational challenges in solving realistic telecommunication network opti-mization problems. CapConFL models some more realistic requirements thathave been ignored by the previous models in the OR literature. However, thereare further interesting extensions of CapConFL that need to be studied andthat are of great relevance for practical applications. For example, CapConFLsimplifies the cost structures in the core network. In a real world setting, cableswith different cost and capacities are available and need to be placed in ductswith sometimes limited capacities. These ducts can be available beforehand,or need to be dug at a much higher cost. Consequently, more complex coststructures could be considered at the core network. If network survivability isan issue, then higher connectivity requirements could be imposed to the corenetwork, or to a subset of its (more important) nodes. These are some of therelevant topics that we believe are of interest for further studies on extensionsof CapConFL.

Acknowledgements The authors thank the two anonymous referees for their valuablecomments and suggestions to improve the paper.

Stefan Gollowitzer was supported by the Fonds de recherche du Quebec - Nature et tech-nologies under grant 163879 within the Programme de Stages Internationaux. Ivana Ljubicwas supported by an APART Fellowship of the Austrian Academy of Sciences (OEAW).This support is gratefully acknowledged.


0% 2% 4% 6% 8% 10%0

5

10

15

20

25

30t

=3

0% 2% 4% 6% 8% 10%0

5

10

15

20

25

30

t=

10

(CutZ) (CutX) (CutZ)+(CutX) Most VI All VI

Fig. 7 Performance chart for 3 minutes (top) and 10 minutes (bottom) runtime


0% 2% 4% 6% 8% 10%0

5

10

15

20

25

30t

=30

0% 2% 4% 6% 8% 10%0

5

10

15

20

25

30

t=

60

(CutZ) (CutX) (CutZ)+(CutX) Most VI All VI

Fig. 8 Performance chart for 30 minutes (top) and 60 minutes (bottom) runtime


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