IP Formulations

In the following we propose some new IP formulations for the considered problems.

IP modeling of BBConFL

We write a natural flow-based integer linear program for the BBConFL problem. For each edge we create a pair of anti-parallel directed arcs, with same length as the original one. Let E~ be the set of these arcs. We denote to the undirected version of an arc

¯

e∈ E~ by e. For an edge e= uv between nodes u and v, we will use a notation (u, v) or (v, u) to explicitly specify the orientation of the corresponding arc ¯e ∈ E. We will~ use the notationeand ¯e, when it is clear from the context, for the sake of compactness.

In our model we will use the following variables: For every ¯e ∈ E~ and client j ∈ D, the variable f_¯e^j indicates if flow from client j uses arc ¯e; for e∈E and access cable k, x^k_e indicates if access cablek is installed on edge e; andz_i indicates if facility iis open or not. We use the notation δ⁺(S) = {(u, v) ∈ E~:u ∈ S, v /∈ S}, δ⁻(S) = δ⁺(V \S), δ(S) ={uv∈E:u∈S, v6∈S}for each S ⊆V and δ⁺(v) =δ⁺({v}) for eachv∈V. We are now ready to provide our integer program.

(IP-4-1) minX

i∈F

µiyi+X

e∈E

ce K

X

k=1

σkx^k_e+MX

e∈E

ceze

X

¯e∈δ⁺(j)

f_¯e^j ≥1 ∀j∈D (4.1)

X

e∈δ¯ ⁺(v)

f_¯e^j = X

e∈δ¯ ⁻(v)

f_e¯^j ∀j∈D, v∈V \(F∪ {j}) (4.2)

X

e∈δ¯ ⁻(i)

f_e¯^j− X

e∈δ¯ ⁺(i)

f_¯e^j ≤yi ∀j∈D, i∈F (4.3)

X

j∈D

dj(f_(u,v)^j +f_(v,u)^j )≤

K

X

k=1

ukx^k_uv ∀ uv∈E (4.4)

Chapter 4. LP-based Approximations and Integrality Gaps 82 yi− X

e∈δ(S)

ze≤0 ∀S⊆V\ {r}:S∩F 6=∅, i∈S (4.5)

yr= 1 (4.6)

f_¯e^j, yi, ze∈ {0,1}

x^k_e non-negative integers

Constraints (4.1) impose that at least one unit of flow leaves the clients. Constraints (4.2) are flow conservation constraints at non-facility nodes. Constraints (4.3) state that the flow only terminates at open facilities. Constraints (4.4) ensure that we install sufficient capacity to support the flow. Finally, constraints (4.5) guarantee that all open facilities are connected to the root via core links, where the constraint (4.6) defines the root facility.

It is not hard to show that an optimal fractional solution for the LP relaxation of IP-4-1 only uses the last cable type with the lowest cost per capacity ratio; see Section 2.1.

This immediately implies the following result.

Remark 4.3. The integrality gap of (IP-4-1) can be arbitrarily large.

The reason why we failed to get an integrality gap ofO(1) for the natural linear program-ming formulation of BBConFL is due to the fact that we have to purchase capacities in discrete unites, however this is not required in DDConFL. Remember that the differ-ences between the BBConFL and DDConFL problems are just due to their considered access cable cost models. In BBConFL, an access cable of type k has a fixed capacity u_k ∈Z>0 and fixed setup costσ_k ∈Z≥0. In DDConFL, an access cable of type khas a setup costσk∈Z≥0, a flow dependent cost of rk∈Z≥0, and unbounded capacity.

We hence focus only on the DDConFL problem. Recall that aρ-approximation algorithm for one problem gives a 2ρ-approximation algorithm for the other; see Section 3.1 for more details.

IP Modeling of DDConFL

We write a flow-based integer linear program for DDConFL. In order to obtain a stronger flow-based formulation, we disaggregate flow variables with respect to cable types. We assume w.l.o.g. that a particular facilityris open and thus it belongs to the core network in the optimal solution and that D∩F = ∅. Also, to simplify the description of our algorithm it will be useful to add an artificialroot client r^∗ with unit demand, connected tor by an edge of 0 length. For every ¯e∈E, cable type~ k∈[K] ={1, . . . , K}and client j ∈ D, the variable f_¯e;k^j indicates if flow from client j uses cable type k on arc ¯e; for e∈E and k∈[K], x^k_e indicates if cable type kis installed on edge e;z_e indicates if the

Chapter 4. LP-based Approximations and Integrality Gaps 83 core cable is installed on edgee; andyiindicates if facilityiis opened. The opening cost C^op, the core installation cost C^core, the fixed installation cost C^fixed and the routing cost C^route of a solution are defined as

C^op =X

i∈F

µ_iy_i; C^core=MX

e∈E

c_ez_e;

C^fixed =

K

X

k=1

C_k^fixed; C^route=

K

X

k=1

C_k^route, where C_k^fixed =σ_kX

e∈E

c_ex^k_e, and C_k^route=r_kX

j∈D

d_jX

¯ e∈E~

c_ef_¯e;k^j , (4.7)

represent the fixed cost and routing cost of the cables of type k, respectively. Given a set of cables I ⊆ [K] and a client j ∈ D, we define the access flow on ¯e ∈ E~ with respect to I and j as f_¯e;I^j = P

k∈If_e;k¯^j ; and the net in-flow on a vertex v ∈ V with respect to I and j, asg_I^j(v) = P

¯

e∈δ⁻(v)f_e;I¯^j −P

¯

e∈δ⁺(v)f_¯e;I^j . We also use the quantity h^j_i = max{g_[K]^j (i),0} for j ∈ D and i ∈ F. Formally, this quantity indicates whether facilityiis serving client j.

With all the notation above, our integer program formulation is as follows.

(IP-4-2) minC^op+C^core+C^fixed+C^route

g^j_[K](j)≤ −1 ∀j∈D (4.8)

g_[K]^j (v) = 0 ∀j∈D, v∈V \(F∪ {j}) (4.9)

g^j_[K](i)≤h^j_i ∀j∈D, i∈F (4.10)

h^j_i ≤yi ∀j∈D,i∈F (4.11)

f_(u,v);k^j +f_(v,u);k^j ≤x^k_uv ∀j∈D, k∈[K], uv∈E (4.12) X

i∈S∩F

h^j_i− X

e∈δ(S)

ze≤0 ∀j∈D, S⊆V\ {r}:S∩F6=∅ (4.13)

yr= 1 (4.14)

g^j_[q,K](v)≤0 ∀j∈D,v∈V \F,1≤q≤K (4.15)

g_[q,K^j _](i)− X

e∈δ(i)

z_e≤0 ∀j∈D, i∈F\ {r},1≤q≤K (4.16)

x^k_e, f_¯e;k^j , y_i, z_e, h^j_i ∈ {0,1} (4.17)

Constraints (4.8)–(4.12) are flow conservation and capacity constraints similar to the ones for the IP-4-1 model. Constraints (4.13) are saying that ifi is the facility serving demand j (that is, the only i for which h^j_i = 1) then for each set S containing i and not containing the root there is a core link connectingS with its complement. In other words, all open facilities are connected to the root via core links, where the constraint (4.14) defines the root facility. Constraints (4.15) and (4.16), called path monotonicity constraints, strengthen the linear relaxation of (IP-4-2) – these constraints ensure that

Chapter 4. LP-based Approximations and Integrality Gaps 84 the cable types along any path used to connect clients to facilities are nondecreasing from each client to its facility. The validity of these constraints follows from the fact that (due to economy of scales) the flow (in the optimal fractional solution) never splits. More precisely, if the flow from a subset of clients shares an edge then it shares the subsequent edges to an open facility; see the paper byGarg et al. (2001) for more details.

We remark that the interesting variables of this IP formulation are (f, x, y, z) = ((f_¯e;k^j ),(x^k_e),(yi),(ze)).

All the other quantities are written in terms of these variables.

The introduction of variables h^j_i may seem artificial, however, in the following section we show that they are needed in order to achieve a constant integrality gap IP.

A naive Model for DDConFL

In this section, we show that an alternative, but perhaps more natural IP formulation for DDConFL has unbounded integrality gap. Consider the following integer programming formulation:

(IP-4-3) min C^op+C^core+C^fixed+C^route

g^j_[K](j)≤ −1 ∀j∈D (4.8)

g^j_[K](v) = 0 ∀j∈D, v∈V \(F∪ {j}) (4.9)

g_[q,K]^j (v)≤0 ∀j∈D,v∈V \F,1≤q≤K (4.15)

g^j_[q,K](i)− X

e∈δ(i)

ze≤0 ∀j∈D, i∈F\ {r},1≤q≤K (4.16)

f_(u,v);k^j +f_(v,u);k^j ≤x^k_uv ∀j∈D, k∈[K], uv∈E (4.12)

yr= 1 (4.14)

g_[K]^j (i)≤y_i ∀j∈D, i∈F (4.18)

y_i− X

e∈δ(S)

z_e≤0 ∀S⊆V\ {r}:S∩F 6=∅, i∈S (4.19)

x^k_e, f_e;k¯^j , y_i, z_e∈ {0,1}

The difference between this formulation and (IP-4-2) is that (IP-4-3) does not have vari-ablesh. We replace constraints (4.10)-(4.11) and (4.13) by (4.18) and (4.19) respectively.

Chapter 4. LP-based Approximations and Integrality Gaps 85 Theorem 4.4. The integrality gap of (IP-4-3) can be arbitrarily large.

Proof. Consider the instance described in Figure 4.1, where the square nodes represent facilities and the circle nodes represent clients. In this instance, K = 1, i.e. we have a unique access cable type, and we set σ1 =r1 = 1. The core cable has a cost (per unit length) equal toM with 1< M < q. For every facilityi∈ {1, . . . , p}, we set an opening cost of µ_i = 1. We also set µ_n=∞. The root facilityr, which must be opened, has an opening cost of 0. The distances are given by the metric completion of the edge costs depicted in the figure, whereL >>1 is a constant larger than any other finite parameter of the instance.

Figure 4.1: An instance of DDConFL withqclients of unit demands and (p+2) potential facilities with facilityras the root node.

The optimal integral solution to this instance can connect all the clients to a fixed facility i^∗ ∈ {1, . . . , p}¹ via access links. Then this open facility is connected to the root node via (unopened) facility nusing core links.

However, the LP relaxation of (IP-4-3) can cheat and open all facilitiesi∈ {1, . . . , p}to the extends of 1/pto serve clients demands. Then it can install core links to the extends of 1/pon the edges connecting them (via noden) to the root node. This means that LP only paysM·L/pfor the core link along edge nr, while the integral solution pays a cost ofM·L on that for the same edge. SinceLwas chosen as an arbitrarily large constant, this is the only relevant value to compare. Hence, the integrality gap is proportional to

p and thus it can be made arbitrarily large.

1Note that they are (almost) collocated.

Chapter 4. LP-based Approximations and Integrality Gaps 86

Im Dokument Network design with facility location (Seite 83-88)