IP Modeling of SHCoFL on Layered Graphs - Network design with facility location

Chapter 5. Complex Connected Facility Location 103

5.1 Preliminaries

Recall that in SHConFL we are given an undirected graphG= (V, E) withV =D∪S,˙ whereDis the set of clients andS is the set of (potential) core nodes. We are also given a set of potential facilitiesF ⊆S and a root r ∈S\F. Furthermore, we are given the costsµi ∈Z≥0 for opening facilityi∈F,aij ∈Z≥0 for assigning clientj ∈Dto facility i∈ F ∪ {r}, and c_e ∈Z≥0 for installing edge e ∈E(S) in the (regional) core network (where E(S) :={uv ∈E :u, v∈S}). Finally, we are given an integer hop limitH ≥1 and an integer connectivity requirementλ≥1.

In SHConFL, we seek for a subset I ⊆ F of facilities to open, a mapping σ(j) : D → I ∪ {r} assigning clients to open facilities, and a subset of edges E⁰ ⊆E(S) such that (S, E⁰) contains, for each facility i∈I, at least λ edge-disjoint (r, i)-paths of length at mostH. The objective is to minimize the total cost

i∈I

µi+X

j∈D

a_σ(j)j+X

e∈E⁰

ce.

The SHConFL problem is obviously NP-hard, as it contains the connected facility loca-tion problem as a special case.

If the set of open facilities is given in advance and the connectivity requirement is fixed to 1 (i.e. λ= 1), then the SHConFL problem reduces to the Hop-Constrained Steiner Tree problem (HST). Manyem (2009) shows that the HST problem is not in APX, even if the edge weights satisfy the triangle inequality. The results by Manyem(2009) immediately imply the following inapproximability result for SHConFL too.

Theorem 5.1. There is no approximation algorithm for SHConFL, even with the edge weights satisfying the triangle inequality, which guarantees a worst case approximation ratio better than Θ(log(|V|)) unlessNP⊆DTIME(n^{log log}ⁿ).

Chapter 5. Complex Connected Facility Location 104

3 2 1

(a)A simple core network where the triangle node represents the root.

4 3 2 1

(b)A hop-expanded layered graph with H = 3 corresponding to facility 4

Figure 5.1: A core graph and a hop-expanded layered graph corresponding to facility 4. Edges in red and blue represent two routing paths between the root and facility 4

in the core graph and the corresponding hop-expanded layered graph.

Multi-Commodity Flow Based Formulation

We formulate SHCoFL on layered graphs. For each facilityi∈F, we choose a commodity and create the following hop-extended directed graph Gⁱ= (Sⁱ, Aⁱ) withH+ 1 layers:

Sⁱ ={r₀} ∪ {u_h:u∈S,1≤h < H u6=r} ∪ {i_H} and Aⁱ ={(u_h−1, vh) :uh−1, vh∈Sⁱ, uv∈ES,1≤h≤H}.

where vh, v ∈S and 0≤ h ≤ H, represents a copy of node v in layer h of Gⁱ. There are two directed edges (uh−1, v_h) and (vh−1, u_h) between layers h−1 andhof Gⁱ corre-sponding to each edge uv in the original graph.

Observe that any (r, i) routing path of length h ≤H in the original (core) graph cor-responds to a directed path of the same length from r₀ to i_h in Gⁱ; see Figure 5.1. In other words, the resulting layered graph Gⁱ= (Sⁱ, Aⁱ) contains only the allowable (r, i) routing paths with respect to the hop constraintH.

Notice that, in the structure described above, we made only the core graph layered and left the rest unchanged.

Chapter 5. Complex Connected Facility Location 105 Now, to model SHCoFL as an IP, we introduce the following binary decision variables:

xij =







1 if clientj is assigned to facility i 0 otherwise

∀i∈F, j∈D;

y_i =







1 if facilityiis opened 0 otherwise

∀i∈F;

zuv =







1 if edgeuv is installed 0 otherwise

∀uv∈E(S).

Additional binary flow variablesf_uv^ih (for commodityi) indicate if a routing path fromr to facilityicontains arc (u, v) in positionh or not.

With all the notation above, our multi-commodity flow-based IP formulation is as fol-lows.

(IP-5-1) minX

i∈F

µ_iy_i+X

j∈D

i∈F∪{r}

a_ijx_ij+ X

e∈E(S)

c_ez_e

i∈F∪{r}

xij= 1 ∀j∈D (5.1)

yi−xij≥0 ∀i∈F, j∈D (5.2)

H−1

h=0

(u_h,i_h+1)∈Aⁱ

f_ui^ih=λyi ∀i∈F (5.3)

(vh−1,u_h)∈Aⁱ

f_vu^i(h−1)− X

(u_h,v_h+1)∈Aⁱ

f_uv^ih= 0 ∀i∈F, u_h∈Sⁱ;u /∈ {r, i} (5.4)

f_rvⁱ⁰≤zrv ∀i∈F, rv∈E(S) (5.5)

H−2

h=1

(f_uv^ih+f_vu^ih)≤zuv ∀i∈F, uv∈E(S), u, v /∈ {r, i} (5.6)

H−1

h=1

f_ui^ih≤zui ∀i∈F, ui∈E(S), u6=r (5.7)

x, y, z, f ∈ {0,1} (5.8)

Constraints (5.1) and (5.2) state that any client has to be assigned to an open facility.

Constraints (5.3) and (5.4) ensureλ units of flow going from each open facility to root r on the layered graphs. This hence guarantees λlength bounded flow paths between r and each open facility in the original graph; thanks to layered graphs. Constraints (5.5)-(5.7) finally guarantee the edge-disjointness of these paths and the correct setting of the edge variables ze.

Chapter 5. Complex Connected Facility Location 106

3 2 1

(a)A core network

Level

1 3

2 4

(b)A corresponding level-expanded core network

Figure 5.2: The level-expanded core network withH = 3 &λ= 2 corresponding to the above solution

Hop-Level Multi-Commodity Flow Based Formulation.

In this section we provide a stronger formulation for SHCoFL. Similar toMahjoub et al.

(2013), we introduce an additional set of variables and constraints to get the distance of each core node from the root node in the solution.

More precisely, given a feasible solution (I^∗, σ^∗, E^∗) to SHConFL, we partition nodes in S into H+ 2 levels according to their distances¹ from r (see Figure5.2) as follows:

• Level 0 only contains r,

• levell, 1≤l≤H, contains nodes with distance lform r,

• levelH+1 contains the nodes that are not connected to r inE^∗.

Then, for every nodeu∈S and level 1≤l≤H+ 1 we introduce an binary variablew^l_u that indicates if nodeu is in level l. We also introduce binary variables a^lk_uv,uv ∈E(S) and 0≤l, k ≤H, indicating if edgeuv belongs to the solution with uin levelland vin levelk, respectively. Remark that |k−l| ≤1.

Let δ(u) = {uv ∈ E(S) : v ∈ S} and let ˆE_S = E_S \δ(r). Constraints (5.9)-(5.20) are devised to transform the core part of a solution over the y and z variables into a

1By distance between two nodes u and v in E^∗ we mean the (minimum) number of edges in E^∗ connectingutov. Note that the maximum possible level for an open facility isH.

Chapter 5. Complex Connected Facility Location 107 corresponding level-expanded solution over the y,w, andavariables.

H+1

l=1

w^l_u= 1 ∀u∈S\ {r} (5.9)

w^H+1_i ≤1−yi ∀i∈F\ {r} (5.10)

w¹_v=a⁰¹_rv=zrv ∀rv∈ES (5.11)

H−1

l=1

a^ll_uv+

H−1

l=1

(a^l(l+1)_uv +a^l(l+1)_vu ) =zuv ∀uv∈EˆS (5.12) a¹¹_uv+a¹²_uv≤w_u¹ ∀uv∈Eˆ_S (5.13) a¹¹_uv+a¹²_vu≤w_v¹ ∀uv∈Eˆ_S (5.14) a^ll_uv+a^l(l+1)_uv +a^(l−1)l_vu ≤w_u^l ∀uv∈EˆS,2≤l≤H−1 (5.15) a^ll_uv+a^l(l+1)_vu +a^(l−1)l_uv ≤w_v^l ∀uv∈EˆS,2≤l≤H−1 (5.16)

a^(H−1)H_vu ≤w_u^H ∀uv∈EˆS (5.17)

a^(H−1)H_uv ≤w_v^H ∀uv∈Eˆ_S (5.18)

w_v^l − X

(u,v)∈δ(v),u6=r

a^(l−1)l_uv ≤0 ∀v∈S,2≤l≤H (5.19)

w, a, y, z∈ {0,1} (5.20)

Constraints (5.9)-(5.10) state that each node should be in exactly one of the possible levels and chosen facilities must be reachable. Constraints (5.11) and (5.12) make the connection between (w, a) variables and variables z. Constraints (5.13)-(5.18) ensure that a variable a^lk_uv can only be 1 if both w_u^l and w^k_v are one. Constraints (5.19) state that a node can only be in levell if it is reached by at least one edge from levell−1.

Recall that, using Constraints (5.9)-(5.20), any binary vector (y, z) defines binary values (y, w, a) in which each node is assigned to a single level. But, analogous toMahjoub et al.

(2013), it can be shown that a fractional (w, a) splits nodes and edges into different levels.

This possibly increases the distance betweenr and open facilities in the level-expanded network induced by (w, a). For example, Figure 5.3(a) illustrates a feasible fractional core network forH = 3 &λ= 2, containing 2 edge-disjoint (fractional) paths of length at most 3 between the root node and each facility which can be specified as follows:

z01 = ˆz34 = 1,zˆ02 = ˆz04 = ˆz12 = ˆz13 = ˆz23 = ˆz24 = 1/2; and Figure 5.3(b) illustrates its corresponding level-expanded solution specified as follows: ¯w₁⁰= ¯w¹₁ = ¯w¹₃ = 2,w¯¹₂ =

w²₂ = ¯w²₄ = ¯w¹₄ = 1/2, ¯a⁰¹₀₁= 1, ¯a⁰¹₀₂= ¯a⁰¹₀₄= ¯a¹²₁₂= ¯a¹²₁₃= ¯a¹²₂₄= ¯a¹²₄₃= ¯a²²₂₃= ¯a²²₃₄= 1/2.

It is easy to see that, unlike the original (fractional) solution, the level-expanded solution violates the required hop constraint; see the routing path (in red) betweenr and facility (with length 5 > H). In fact this is the main reason why we are interested in this level based transformation as we are able now to cut away such fractional solutions by

Chapter 5. Complex Connected Facility Location 108

3 2 1

(a)

Level

1 2 4

2 3 4

(b)

Figure 5.3: A feasible fractional core network withH= 3 &λ= 2 and its corresponding level-expanded solution.

ensuring the existence of the hop-limited arc-disjoint routing paths betweenr and open facilities in the level-expanded network, instead of the original network. In the following we model this.

We formulate this on layered graphs to ensure the hop constraints. For each facility i∈F, we create a directed level- and hop-expanded graph Gⁱ_L= (S_Lⁱ, Aⁱ_L) with

S_Lⁱ ={r₀⁰}

∪ {u^l_h:u∈S,1≤l≤h≤H−1, u6=r}

∪ {i^l_H : 1≤l≤H},and

Aⁱ_L={(u^l_h−1, v_h^k) :u^l_h−1, v_h^k∈S_Lⁱ, uv∈E_S,1≤h≤H,|l−k| ≤1}

∪ {(i^l_h, i^l_h+1) : 1≤l≤h≤H−1}

In which the copy of node u in layer h and level l is denoted by u^l_h, while (u^l_h−1, v_h^k) denotes the directed arc corresponding to the copy of edgeuvwith nodeuin layerh−1 and levell and nodev in layer h and levelk; see Figure5.4.

Now, corresponding to each arc of the level- and hop-expanded layered graph, we intro-duce a binary flow variableg^ihlk_uv indicating that a path fromrto facilityiuses arc (u, v) in hop h whereu is at level land v is at level k.

With all the notation above, our extended IP formulation for SHCoFL is as follows.

Chapter 5. Complex Connected Facility Location 109

4 3 2 1

Figure 5.4: The level- and hop-expanded layered graph with H = 3 & i = 4 corre-sponding to the core graph shown in Figure5.1a.

(IP-5-2) minX

i∈F

µ_iy_i+X

j∈D

i∈F∪{r}

a_ijx_ij+ X

e∈ES

c_ez_e

(5.1)−(5.2) (5.9)−(5.19)

l=1

(v_H−1^k ,i^l_H)∈Aⁱ_L

g^i(H−1)kl_vi =λyi ∀i∈F (5.21)

(v_h−1^k ,u^l_h)∈Aⁱ_L

g^i(h−1)kl_vu −X

(u^l_h,v_h+1^k )∈Aⁱ_L

g_uv^ihlk= 0 ∀i∈F, u^l_h∈S_Lⁱ, u6=r, h≤H−1 (5.22)

gⁱ⁰⁰¹_rv ≤a⁰¹_rv ∀i∈F, rv∈E_S (5.23)

H−2

h=l

(g_uv^ihll+g^ihll_vu )≤a^ll_uv ∀i∈F, uv∈EˆS\δ(i),1≤l≤H−2 (5.24)

H−2

h=l

g_uv^ihl(l+1)+

H−2

h=l+1

g_vu^ih(l+1)l≤a^l(l+1)_uv ∀i∈F, uv∈Eˆ_S\δ(i),1≤l≤H−2 (5.25)

H−1

h=l

g^ihll_ui ≤a^ll_ui ∀i∈F, ui∈δ(i)\δ(r),1≤l≤H−1 (5.26)

H−1

h=l

g_ui^ihl(l+1)≤a^l(l+1)_ui ∀i∈F, ui∈δ(i)\δ(r),1≤l≤H−1 (5.27)

w, a, y, x, z, g∈ {0,1}

Constraints (5.21)-(5.27) ensure the existence of the λ arc-disjoint paths between the root node and each open facility in the layer- and level-extended graphs. More precisely, Constraints (5.21)-(5.22) guarantee that at leastλunits of flow go fromr to each open

Chapter 5. Complex Connected Facility Location 110 facility on the the layer- and level-extended graphs. Constraints (5.23)-(5.27) then link the flow to thea variables.

LetP1 and P2 denote the feasible space of the LP relaxation corresponding to formula-tions IP-5-1 and IP-5-2, respectively.

Theorem 5.2. P roj_X,Y,Z(P₂)⊆P roj_X,Y,Z(P₁).

Proof. Given a fractional solution (¯x,y,¯¯ z,¯a,w,¯ ¯g) to IP-5-2. We construct an equivalent solution (ˆx,ˆy,ˆz,ˆf) to IP-5-1 as follows. We take the vectors ˆx= ¯x,yˆ =y, and¯ ˆz =¯z the same for the newly constructed solution. But for the flow variables we set

fˆ_uv^ih=X

l,k

g^ihlk_uv , ∀i∈F, uv∈E(S),0≤h≤H.

It is not hard to check that the constructed solution satisfies Constraints (5.1)-(5.8), and hence is a feasible solution to IP-5-1. This completes the proof.

Recalling the instance described in Figure5.3, we conclude that IP-5-2 is stronger than IP-5-1 in terms of the lower bounds provided by their linear relaxations.

Im Dokument Network design with facility location (Seite 105-112)