The Two Level Network Design Problem with Secondary Hop

The Two Level Network Design Problem with Secondary Hop Constraints Stefan Gollowitzer1 , Lu´ıs Gouveia2, and Ivana Ljubi´c3 1

2

Department of Statistics and Operations Research, University of Vienna, Austria [email protected] Departamento de Estat´ıstica e Investigac¸a˜ o Operacional - Centro de Investigac¸a˜ o Operacional, Faculdade de Ciˆenc¸ias, Universidade de Lisboa, Portugal [email protected] 3 Institute of Computer Graphics and Algorithms, Vienna University of Technology, Austria [email protected]

Abstract. The Two Level Network Design problem asks for a cost-minimal Steiner subtree of a given graph G = (V, E) that connects all primary customers using a primary technology only, and all secondary customers using either the primary or the secondary technology. Thereby, the secondary technology is cheaper but less reliable and hence, hop constraints on the length of each secondary path are imposed. In addition, in some applications facility opening costs need to be paid for transition nodes, i.e., for nodes where the change of technology takes place. We consider various MIP models for this new problem and derive a new class of strong inequalities that we call generalized cut-jump constraints. We also show that these inequalities can be obtained by projecting the cut-set formulation obtained on a graph in which we split the potential facility locations and introduce layers for installing the secondary technology.

1 Introduction Two Level Network Design (TLND) problems with a tree structure arise when local broadband access networks are planned in areas, where no existing infrastructure can be used, i.e., in so-called greenfield deployments (see, e.g., [1]). Topological network design for mixed strategies of Fiber-To-The-Home and Fiber-To-The-Curb, i.e., some customers are served by copper cables, some by fiber optic lines, can be modeled by an extension of the TLND. Since the secondary technology (e.g., copper) is usually much cheaper but also less reliable than the primary technology (e.g. optical fiber), hop-constraints on the length of secondary paths need to be introduced. Definition 1 (TLNDSH). We are given an undirected graph G = (V, E) with a root r and a set of customers R ⊆ V \ {r}. To each edge e ∈ E, two installation costs c1e ≥ c2e ≥ 0 are associated. These correspond to the primary and secondary technology, respectively. The primary edges are more reliable, and hence, more expensive. The set of customers, R, is partitioned into two subsets P and S. We are also given facility opening costs fi ≥ 0∀i ∈ V , and a hop limit H. Our goal is to determine a cost-minimal subtree of G satisfying the following properties: J. Pahl, T. Reiners, and S. Voß (Eds.): INOC 2011, LNCS 6701, pp. 71–76, 2011. c Springer-Verlag Berlin Heidelberg 2011 

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(Psub) The subgraph made of primary edges is a tree rooted at r interconnecting the primary nodes in P. (S) Each secondary node in S is connected to the root by a path consisting of primary and/or secondary edges. (F) Facility opening costs are payed for transition nodes i ∈ V , i.e., nodes where change of technology takes place, (H) for each secondary node v, the number of secondary edges on the path between v and r may not exceed H, and (E) on each edge e ∈ E at most one of technologies is installed. We observe that, since the edge costs are non-negative, there always exists an optimal solution which is a Steiner tree interconnecting the nodes from R ∪ {r}. Furthermore, since c1e − c2e ≥ 0, this Steiner tree is composed of a subtree of primary edges (primary subtree) and a union of subtrees of secondary edges (secondary subtrees). Each secondary subtree is rooted in a node of the primary subtree. Due to facility opening costs and hop-constraints, every node from V may be a leaf of the primary subtree. The TLND with secondary hop constraints has not yet been studied in the literature. Minimum spanning trees with two technologies and secondary distance constraints have been introduced in [6]. In [4] the TLND is considered with weighted hop constraints defined as follows: the goal is to construct a two level minimum spanning tree such that for each node k, the r-k path contains a weighted number of primary and secondary edges (with weights w1 and w2 , respectively) which does not exceed H (for given w1 , w2 , H ∈ N).

2 MIP Models There are two challenges in modelling the TLNDSH problem: a) facility nodes and b) hop constraints. MIP formulations focused on modelling facility nodes are studied in in [3]. In this short abstract, we mainly concentrate on modeling the hop constraints on secondary trees. The models will be derived on graph G = (V, A) with the set of directed arcs A = {i j | {i, j} ∈ E, j = r} and with ckij = ckji = cke for k = 1, 2 and e = {i, j}. We use sets IH := {1, 2, . . . , H} and IH0 := IH ∪ {0}. For any W ⊂ V we denote its complement set by W c = V \ W . For any M, N ⊂ V , M ∩ N = 0, / we denote the induced cut set of arcs by (M, N) = {i j ∈ A | i ∈ M, j ∈ N}. In particular, let δ − (W ) = (W c ,W ), δ + (W ) = (W,W c ) and δ − (i) = (V \ {i}, {i}). The following binary variables are used in our models: Variables x1i j (x2i j ) will indicate if the primary (secondary) technology is installed on i j ∈ A, while zi will be one if a facility is installed on i ∈ V . For a set of ˆ = ∑ ˆ x , for  = 1, 2, and (x1 + x2 )(A) ˆ = ∑ ˆ x1 + x2 . arcs Aˆ ⊆ A, we write x (A) ij i j∈A i j i j∈A i j For W ⊆ V we define z(W ) = ∑i∈W zi . A Cut-Jump Model. The first formulation uses a generalization of the well known jump constraints that are crucial for modelling problems with network design constraints [2]. Let [S0 , S1 , . . . , SH+1 ] be a partition of V , such that the root node r ∈ S0 and SH+1 ∩ S = 0/ (observe that some of sets Si , for i ∈ {1, . . . , H} may also be empty). We call JH =  J(S0 , S1 , . . . , SH+1 ) = (i, j):i< j−1 [Si , S j ] where [Si , S j ] = {uv ∈ A : u ∈ Si , v ∈ S j } a Hjump. In fact, without loss of generality, we can consider only SH+1 = {i} for i ∈ S.

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Letting JH denote the set of all possible H-jumps, we derive the following formulation for the TLNDSH: (JUMP)

∑ (c1i j x1i j + c2i j x2i j )+ ∑ fi zi

min

i∈V

i j∈A

+

x (JH ) + x (δ (S0 )) ≥ 1 2

1

zj +

∑

i j∈δ − ( j),i=k

x2i j

≥ x2jk

x1 (δ − (W )) ≥ 1 −

x (δ (W )) ≥ z j 1

−

(x + x )(δ (i)) ≤ 1 1

2

x1i j , x2i j , zi

∀JH : {r} ∪ P ⊆ S0 , SH+1 = {i}, i ∈ S

(1)

∀ jk ∈ A : k ∈ P

(2)

∀W ⊆ V \ {r}, W ∩ P = 0/

(3)

∀W ⊆ V \ {r} : j ∈ W

(4)

∀i ∈ V

(5)

∈ {0, 1} ∀i j ∈ A, ∀i ∈ V

(6)

The new cut-jump constraints (1) state that each secondary customer is connected to the root by primary or secondary edges, whereas the length of secondary edges along the path does not exceed H. Lemma 1. Formulation JUMP is valid for the TLNDSH. Proof. To show that the property (S) is satisfied, assume that there exists a secondary customer i that is not connected to r. Let C be the set of nodes belonging to the same connected component as i. We now set SH+1 = {i}, SH = C, S0 = V \ (C ∪ {i}), and / for h ∈ {1, . . . , H − 1}. Then constraint (1) is violated, which is a contradiction. Sh = 0, Constraints (5) ensure that each node has an in-degree of at most 1. This guarantees that no two opposing arcs are installed at the same time, and that a primary and a secondary arc cannot enter the same node. Inequalities (2) are an adaptation of degree-inequalities proposed by [7] for the Steiner tree problem. They force opening a facility in each node whose secondary out-degree is positive, and there are no ingoing secondary arcs. Hence, together with (5) and (1), (2) ensure that the property (F) is fulfilled. We now show property (Psub), i.e., that the primary subnetwork is a subtree rooted in r. Due to the cut set (3), primary customers are connected to the root using primary arcs only. Assume a solution contains an infeasible path. The case of two or more subpaths of each technology is impossible due to connectivity cuts (4). So the path consists of two primary subpaths connected by a secondary subpath: But then inequality (1) is violated for the following case: Nodes in the primary subpath containing the root node are in S0 ; the nodes of the secondary subpath are in set S1 ; all but the last node in the remaining primary subpath are in S2 and the last node is in SH+1 . Finally, assume that there exists a binary solution (x1 , x2 , z) not satisfying property (H). That is, there is a customer i ∈ S which is connected to r by a path in which more than H secondary arcs are used. Without loss of generality we can assume that there are exactly H + 1 arcs in this path. Denote the nodes on the secondary path as v1 , . . . , vH+2 . Let S0 denote the set of nodes on that path that are adjacent to primary arcs only. We now consider the following jump-set: JH = {S0 , {v1 }, . . . , {vH+1 }, {vH+2 }}. Obviously, for JH defined in this way, constraints (1) are violated, which is a contradiction. 

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3 Layered Graph Models for TLNDSH In this section we show how to model the TLNDSH as a Steiner Arborescence problem with additional degree constraints in an adequate layered graph. Similar ideas have been proposed for the hop-constrained spanning tree problem [5] and the hop-constrained connected facility location problem [8]. We first consider a layered graph model without node splitting and then a layered graph model with node splitting. 3.1 Solving the TLNDSH as a Steiner Arborescence Problem with Facility and Node-Degree Constraints A first layered graph GL = (V L , AL ) with the set of terminals denoted by RL is constructed as follows: V L :=

H 

V h where

AL :=

h=0

H 

Ah where

h=0

V := {i | i ∈ V },

A := {i0 j0 | i j ∈ A},

V h := {ih | i ∈ V \ P} ∀h ∈ IH−1 ,

Ah := {ih−1 jh | i j ∈ A, j ∈ P} ∀h ∈ IH−1 ,

V H := {iH | i ∈ S},

0 AH := {ih iH | i ∈ S, h ∈ IH−1 }

0

0

0

RL :={i0 ∈ V0 | i ∈ P} ∪V H and

∪ {iH−1 jH | jH ∈ V H , i j ∈ A}.

The costs of the arcs are set as follows: i) arcs in A0 are assigned the primary edge costs; ii) arcs in Ah for h ≥ 1 are assigned secondary edge costs, with only exception of the 0 arcs of type (ih , iH ), h ∈ IH−1 , that are assigned a cost of zero. Figure 1a) illustrates a small segment taken from a layered graph obtained this way. We associate binary variables to the arcs in AL as follows: variable x1i j to arc i j ∈ A0 , h−1 j h ∈ Ah ∀h ∈ I , and variable x2h to arc variable zi to node i ∈ V 0 , variable x2h H i j to arc i ii ih iH ∈ AH . Let δ − (W ) = (V L \ W ;W ) denote a directed cut in GL and X(δ − (W )) the sum of all x-variables corresponding to arcs in that cut. Then, the model reads: H

(SAz )

min

∑ c1i j x1i j + ∑ ∑ c2i j x2h i j + ∑ f i zi

i j∈A

s.t.

i j∈A h=1

−

X(δ (W )) ≥ 1 zi ≥ x21 ij

∑−

i j∈δ (i)

H

(x1i j + ∑ x2h ij ) ≤ 1

i∈V

∀W ⊆ V L \ {r},W ∩ RL = 0/

(7)

∀i j ∈ A

(8)

0 1

1

∀j ∈V

(9)

h=1

x1i j , x2h i j , zi ∈ {0, 1}

∀i j ∈ A, h ∈ IH0 , ∀i ∈ V

(10)

Consider an optimal binary solution of the SAz formulation. Removing the arcs in AH and ignoring the indices h of the remaining arcs, one obtains a two-level Steiner tree with secondary depth less or equal than H in the original graph and with the cost obtained as a sum of facility opening, primary and secondary edge costs. In other

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words, variables x2h can be projected into the original variable space as follows: x2i j = 2h 2h ∑H h=1 xi j ∀i j ∈ A, while the variables xii are ignored. Lemma 2.

υLP (SAz ) ≥ υLP (JUMP).

Proof. We prove the claimed relation by showing that an LP-optimal solution (x, z) of SAz can be projected onto an LP-feasible solution (¯x, z¯ ) of formulation JUMP: For 0 an LP-optimal solution zi = max j x21 i j holds ∀i ∈ V . Together with inequalities (7) this implies  x1 − 0 X(δ (i )) ≥ i21j ∀i0 ∈ V 0 and (11) xi j X(δ − (ih )) ≥ x2,h+1 ij

∀ih ∈ V h , h ∈ IH−1 .

(12)

Then inequalities (7) imply constraints (4) and (3). By summing up (12) and (8) we derive (2). Inequalities (9) imply (5). (8) together with (12) guarantee (4). Finally each constraint in (1) is the projection of constraint (7) where W = {i0 | i ∈ S0c } ∪ {ik | i ∈

H 

Sk , k ∈ IH } ⊂ V L \ {r}.



k=h

3.2 Generalized Cut-Jump Inequalities We describe how to eliminate facility constraints (8) from the Steiner arborescence formulation in order to obtain even a stronger formulation. We do so by splitting the nodes in V 0 into one primary and one facility copy and obtain a new layered graph G¯ L = (V¯ L , A¯ L ). The set of nodes in G¯ L is then V¯ L = V L ∪ V z where V z = {iz | i ∈ V }. We adapt the sets of arcs in GL by replacing A1 by A¯ 1 := {iz j1 | i j ∈ A} and adding Az := {i0 iz | i ∈ V }. The costs of arcs in Ah , for h ∈ IH \ {1} are set as above. Arcs of Az are split facility nodes and therefore ci0 iz := fi ∀i ∈ V . Figure 1 illustrates this transformation. Associating binary variables zi to arc i0 iz ∈ Az and using the variables introduced for formulation SAz we can model the TLNDSH as Steiner Arborescence problem with node degree constraints on G¯ L . We denote it by SA. In the proof of Lemma 2 we have seen that the cut-jump inequalities are a special case of Steiner cuts in the layered graph GL . We now show how a more general family of stronger inequalities can be derived from Steiner cuts in the split layered graph G¯ L . Looking at the valid Steiner cuts in G¯ L we can derive two different types of generalized cut-jump constraints (GCJ) as follows: a) Let S0 be a superset of S0 . Then, the following cuts are valid: x1 (δ + (S0 )) + x2 (J) + z(S0 \ S0 ) ≥ 1 ∀{r} ∪ P ⊆ S0 ⊂ S0 , SH+1 = {i}, i ∈ S, i ∈ S0 (13)

b) Let S0 be a subset of S0 . Then the following cuts are valid for TLNDSH: x1 (δ + (S0 )) + x2 (J) ≥ 1 ∀{r} ∪ P ⊆ S0 , SH+1 = {i}, i ∈ S, S0 ⊂ S0

(14)

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S. Gollowitzer, L. Gouveia, and I. Ljubi´c

10

a)

20

30

20

30

21

31

2z

3z

32

21

31

10

b)

32

Fig. 1. Illustration of the transformation from layered graph GL to G¯ L for H = 2. Node 1 is a primary customer, thus the copy indexed by 0 is a terminal. Node 2 is a Steiner node and node 3 is a secondary customer, thus the copy with index 2 is a terminal as well. Dotted arcs are the ones in AH with a cost of 0.

In our future work we intend to examine the (generalized) cut-jump inequalities more completely, in terms of studying their practical usefulness as well as in finding a suitable and intuitive interpretation for them. We also plan to implement a branch-and-cut algorithm involving the (generalized) cut-jump inequalities and to investigate the practical importance of the layered graph models when compared to other, weaker formulations. Acknowledgement The authors like to thank the referees for their useful comments and suggestions. Luis Gouveia was funded by Centro de Investigac¸a˜ o Operacional [MATH-LVT-Lisboa-152].

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