Multicommodity flow models for spanning trees with hop constraints

E L S E V I E R European Journal of Operational Research 95 (1996) 178-190

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

T h e o r y a n d M e t h o d o l o g y

Multicommodity flow models for spanning trees with hop constraints

L u i s G o u v e i a

DEIO - CIO Faculdade de Ci~ncias da Universidade de Lisboa Bloco C / 2 - Campo Grande CIDADE UNIVERSITARIA 1700 Lisboa, Portugal

Received 6 June 1994; revised 22 February 1995

Abstract

In this paper we compare the linear programming relaxations of undirected and directed multicommodity flow formulations for the terminal layout problem with hop constraints. Hop constraints limit the number of hops (links) between the computer center and any terminal in the network. These constraints model delay constraints since a smaller number of hops decreases the maximum delay transmission time in the network. They also model reliability constraints because with a smaller number of hops there is a lower route loss probability. Hop constraints are easily modelled with the variables involved in multicommodity flow formulations. We give some empirical evidence showing that the linear programming relaxation of such formulations give sharp lower bounds for this hop constrained network design problem. On the other hand, these formulations lead to very large linear programming models. Therefore, for bounding purposes we also derive several lagrangean based procedures from a directed multicommodity flow formulation and present some computational results taken from a set of instances with up to 40 nodes.

Keywords: Integer programming; Linear programming relaxations; Multicommodity flows; Trees; Hop constraints; Reliability

1. Introduct ion

The terminal layout problem (TLP) consists of f inding the best way to link n terminals, at different locations, to a central node (a computer site or a concentrator). The optimal topology for this type of problems, corresponds to a tree in a graph G = (V, E) with all but one of the vertices in V corresponding to the terminals. The remaining node, the root node, refers to the computer site (or the concentrator) and edges in E correspond to the feasible wiring.

In this paper we discuss hop constraints in the context of the TLP. These constraints l imit the number of hops (links) between the root and any terminal and are closely related with the max imum delay

transmission time between any terminal and the computer center. Notice that the transmission time in any link of these networks is almost negligible and as a consequence, the maximum delay associated to a message in any path is directly proportional to the number of hops in that path.

LeBlanc and Reddoch (1990) have pointed out that hop constraints can also model reliabili ty constraints when designing telecommunication networks. Assume that we associate a relability a to each link of the network, i.e., the reliabili ty measures the probabi l i ty that the link will be operational. One is interested in guaranteeing that each message sent from the computer site to each terminal has, at least, a certain probabi l i ty /3 o f reaching its destination.

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L. Gouveia / European Journal of"Operational Research 95 (1996) 178-190 179

This corresponds to guaranteeing that the reliability associated to each path from the central node to any terminal is not less than /3. Under the assumption that the event "failure of link (i, j ) " is independent of what happens to any other link, the reliability associated to each path is given by a k where k is the number of links in that path. As soon as the parameter k exceeds a certain limit, the value a k becomes less than /3. Thus, these reliability require- ments can be modelled as hop constraints. As an example, let a = 0.95 and /3 = 0.80. As a 4 ~_>/3 and a 5 </3, no more than four hops can be included in each path starting from the root. This example also indicates that from the reliability point of view it is important to set a small value for the maximum allowed number of hops,

Additionally, hop constraints can be used to avoid degradation of signal quality and in many networks, voice paths are constrained to two hops while data paths are constrained to three hops (LeBlanc and Reddoch, 1990). Hop constraines were already discussed by Balakrishnan and Altinkemer (1992) as a means for generating alternative and good base solutions that span a wide range of cost and service levels for a more general model.

Consider, then, the following graph theoretical problem which will be denoted as the Hop Con- strained Minimal Spanning Tree (HMST) problem:

Consider a graph G = (V, E) where V = 0, 1 . . . . . n, with costs cij for each edge (i, j ) ~ E and a natural number H. We want t o f ind the directed minimal spanning tree such that eac h path from node 0 to any other node has no more than H hops.

As pointed out in Gouveia (1995), the HMST with H = 2 is equivalent to a version of the Simple Uncapacitated Facility Location (SUFL) problem where the potential facility sites coincide with the locations of the clients to be served. It is well known that the SUFL problem is NP-hard which implies that the HMST is also NP-Hard. Therefore, we focus only on instances with H _> 3 because the HMST problem with H = 2 is widely studied in the location theory literature. Lower bounding schemes for the HMST were already proposed in Gouveia (1995). The reported results indicate that the formulations presented in that paper have to. be considerably strengthened in order to reduce the reported gaps.

It is well known that for a wide class of network design problems, the linear programming (LP) relaxations of multicommodity flow formulations give, in general, bounds of good quality. In the present paper we discuss multicommodity flow formulations for the HMST problem. We start by introducing an undirected multicommodity flow formulation for the HMST which involves O(n 3) variables and O(n 4) constraints. By replacing edges variables with arc variables (i.e., duplicating the number of variables) we show how to derive a more compact multicommodity flow formulation which uses only O(n 3) constraints and has the same LP value as the undirected formulation. In many cases, the directed formulation still produces fairly large LP models even for small values of n. We present a simple arc elimination test which considerably reduces the size of the corresponding LP models. However, the reduced models are still too large for many of the cases. Therefore, we also derive two lagrangean relaxations from the directed model. One advantage associated to one of the proposed relaxations is that the relaxed problem does not satisfy the integrality property, which means that the bounds given by such a relaxation may be better than the bounds given by the corresponding LP relaxation. The reported computational results show that in many cases, we are able to close the duality gap. We present liftings of the hop constraints which lead to sharper models and their inclusion in the directed multicommodity flow model produce improvements on the value of the original lower bounds. Significant improvements are obtained for instances with a small value of H.

2. An undirected multicommodity flow formulation

Consider the binary variables X(;j) (i = 0 . . . . . n - 1; j = i + 1 . . . . . n) such that X(i:) = 1 iff edge {i, j} is in the minimal spanning tree. Consider also the directed binary variables Yijk (i = 0 . . . . . n; j , k = 1 . . . . . n; i 4: k; i 4: j ) which specify whether edge {i, j} is used in the direction from i to j in the path from the root to node k. Then, we have the following undirected formulation for the HMST problem:

180 L. Gouveia /European Journal of Operational Research 95 (1996) 178-190

Formulation UMCF

m i n k k cijX(ij) i=O j=i+l

subject to:

n--I k E X(ij)=Fl; i=0 j=i+l

i=0 i=1

k Y~jj = 1 i=0

rojk -< X(oj) Y,j + r,.ih <- x<i+)

k # i , h # j ;

k k Y + j k < H i=0 j=l

rijk E {0, 1}

x.j) {o, 1}

(2.1)

(2.2)

j , k = l . . . . . n , j # k ;

(2.3a)

j = 1 . . . . . n; (2.3b)

j , k = 1 . . . . . n; (2.4a)

i , j , k , h = l . . . . . n,

k = l , . . . , n .

i = 0 . . . . . n, j , k = l . . . . . n;

(2.4b)

(2.5)

(2.6) i = 0 . . . . . n, j = i + l . . . . . n.

(2.7)

To simplify the indexing, we have not considered variables Ykik (i, k = 1 . . . . . n, i 4 k). Without constraints (2.5) the above formulation is exact (in the sense that the optimal solution of its LP relaxation is always integer) for the minimal spanning tree problem (Martin, 1986). Constraints (2.4b) are bidirectional forcing constraints introduced by Balakrishnan et al. (1989) for the uncapacitated network design problem and state that if a given edge {i, j} is used in the path from the root to any two given nodes h and k then it is used in the same direction (either from i to j or from j to i) in both paths. Constraints (2.5) are the hop constraints and state that no more than H edges are included in the unique path between the root and node k. In the following, let PL denote the LP relaxation of formulation P. To define UMCF L we replace constraints (2.6) by the set of constraints Yijk > 0 , i = O . . . . . n, j , k = l . . . . . n (denoted by (2.6') in' the sequel) and replace constraints (2.7) by the set O<__X(ij)<_I , i = 0 . . . . . n; j = i + l . . . . . n (denoted by (2.7') in the sequel). Notice that when defining UMCF L we need not include the upper

bound constraints Y;jk < 1, i = 0 . . . . . n, j , k = 1 . . . . . n because they are implied by (2.4a), (2.4b) and (2.7').

The UMCF model involves too many (O(n4)) constraints, which implies that the corresponding LP relaxation can only be solved for very small instances. To reduce the number of forcing constrains associated to the model we may replace constraints (2.4b) by the weaker set of unidirectional forcing constraints:

Yijk ~ X<ij) and Yjik ~ X<ij)

i , k = 1 . . . . . n, j = i + 1 . . . . . n. (2.4c)

In the sequel, the model UMCF with constraints (2.4c) instead of (2.4b) is denoted by Weak UMCF (WUMCF). The WUMCF model is still valid for the HMST and involves only O ( n 3) constraints. How- ever, some results given for several network design models not involving hop constraints (Balakrishnan et al., 1989, 1994) indicate that the corresponding LP models get much stronger when the traditional unidirectional forcing constraints are replaced by the bidirectional forcing constraints. In Section 9 we present some results taken from graphs with n = 15 which show that a similar situation is maintained in our hop constrained model. A better alternative to replacing the set (2.4b) by the weaker set (2.4c) consists of reformulating the HMST as a directed model. In the next two sections we show that it is possible to derive a directed formulation for the HMST with the same optimal LP value as the original UMCF model and nearly the same number of constraints as the more compact and weaker WUMCF model.

3. A directed multicommodity flow formulation

An undirected HMST instance can be transformed into an equivalent directed HMST model if we replace each edge {i, j} of the undirected graph with two directed arcs (i, j ) and ( j , i) and the costs of these two arcs is equal to the cost of the original edge. We also assume that the arcs are directed outward from the root and any edge {0, i}, i = 1 . . . . . n, is replaced by only one arc (0, i). To derive a directed multicommodity flow formulation for the HMST consider the binary directed variables Xij ( i = 0 . . . . . n, j = l . . . . . n, i # j ) such that X i j = l

L. Gouveia / European Journal of Operational Research 95 (1996) 178-190 181

iff arc (i, fl is in the minimal spanning tree. As with the undirected model, we also consider the directed binary variables Yij* which specify Whether arc ( i , j ) is used in the path from the root to node k. As the variables Y~jk are used in both models, the directed model also includes constraints (2.3a), (2.3b), (2.5) and (2.6) from the undirected model which for con- sistency, are renamed (3.3a), (3.3b), (3.5) and (3.6) for the directed model. Then, we have the following directed formulation for the HMST problem:

F o r m u l a t i o n D M C F

min ~ ~ cijXij (3.1) i = 0 j = l

subject to: n

E Xij = 1 j = 1 . . . . . n; (3.2) i = 0

, ( 3 . 3 a )

, ( 3 . 3 b )

, ( 3 . 5 )

, ( 3 . 6 ) ;

Yijk <~Xij i = 0 . . . . . n, j , k = l . . . . . n; (3.4)

Xije{o,a } i = 0 . . . . . n, j = 1 . . . . . n. (3.7)

To simplify the indexing, we have not considered variables Xii and Yiik (i = 1 . . . . . n). The same also happens with variables Ykik (i, k = 1 . . . . . n, i 4= k). To define DMCF L we replace constraints (3,6) by the set (2.6') (denoted by (3.6') in the sequel) and replace (3.7) by Xij >_ O, i = 0 . . . . . n, j = 1 . . . . . n (denoted by (3.7') in the sequel). Notice that when defining DMCF L we need not to include the upper bound constraints Xij < 1, i = 0 . . . . . n, j = 1 . . . . . n, because they are implied by constraints (3.2).

One consequence of a result given by Maculan (1986) is that the projection of the polyhedron defined by (3.2), (3.3a), (3.3b), (3.4), (3.6') and (3.7') into the space of the Xij variables is described by (3.2), (3.7') and the directed subtour elimination constraints

E E X i j ~-~[S[- 1 i~S jES

VS___{1 . . . . . n} and [SI>_ 2. (3.8)

Notice that when S = {i, j}, constraints (3.8) can be rewritten as Xij-~-Xji ~__ 1. This will be used in the proof of Result 1 presented in the next section. The polyhedron defined by (3.2), (3.8) and (3.7') gives a complete description of the convex hull of the incidence vectors of directed minimal spanning trees (Edmonds, 1968). This result will be used in Section 7 for tightening two lagrangean relaxation schemes derived directly from the DMCF formulation.

In the next section we show that the LP relaxations of the two models, DMCF and UMCF, are equivalent, (See Goemans and Myung (1993) and Balakrishnan et al. (1994) for similar equivalence results involving other non-trivial problems related with trees.)

4. E q u i v a l e n c e o f U M C F L a n d D M C F L

Let F ( Q ) denote the set of feasible solutions of a given formulation Q. In this section we shall show that

F(UMCFrL) = { ( X(ij), Yijk ):

there exists X;j such that X(ij) -- Xiy + Xji

×and (Xij, Yijk) ~ F(DMCFL) }.

(4.1)

The equality (4.1) states that if we add the equali- ties

X(0j) = X0j j = 1 . . . . . n, (4.2a)

X(ij)=Sij-~-Xji i = 1 . . . . . n, j = i + l . . . . . n

(4.2b)

to the polytope F(DMCF L) and project out the directed variables X~j we obtain the polytope F(UMCFL).

Result 1. (4.1) is valid.

Proofi Assume that we have an undirected solution (X(i j) , Yijk ) and assume there exists a directed solution (X i j , Yij k) feasible for DMCF L and such that (4.2a) and (4.2b) hold. Next we show that the solution (X(ij), Yijk) is feasible for UMCF L. Clearly, the

182 L. Gouveia / European Journal of'Operational Research 95 (1996) 178-190

Yijk variables satisfy constraints (2.3a), (2.3b), (2.5) and (2.6') in the undirected model because they satisfy constraints (3.3a), (3.3b), (3.5) and (3.6') in the directed model. It remains to show that the undirected variables X(ij) satisfy (2.2), (2.4a), (2.4b) and (2.7'). By summing up constraints (3.2) and using (4.2a) and (4.2b) we obtain constraints (2.2). A given bidirectional forcing constraint (2.4b) for (i, j, k, h) is implied by two directed forcing constraints (3.4), one for the triple (i, j, k) and the other for the triple ( j , i, h) which shows that (2.4b) are implied by (3.4). The unidirectional forcing constraints (2.4a) are implied by the corresponding directed forcing constraints (3.4) for i = 0. Finally, notice that the directed solution satisfies the constraints Xi j + Xji < 1 ( i , j = 1 . . . . . n) which are the directed subtour elimination constraints (3.8) for sets of two nodes. These inequalities together with (4.2a), (4.2b) and (3.7') imply that the undirected solution satisfies (2.7'). This proves one of the inclusions implicit in (4.1).

Conversely , consider a feasible solution ( X ( u ) , Y u k ) for UMCF L. To see that there exists a directed solution ( X u, Yuk) feasible for DMCF L and such that (4.2a) and (4.2b) hold, we define the value of the directed X u variables as

X0j = X(0j) j = 1 . . . . . n, (4.3a)

Xi j = Mi j i = 1 . . . . . n - l , j = i + l . . . . . n ,

(4.3b)

X i j = X ( j i ) - Xji i = 2 . . . . . n , j = 1 . . . . . i -- 1,

(4.3c)

where Mij = max{Yij k I k = 1 . . . . . n, k # i} for i = 1 . . . . . n - 1, j = i + 1 . . . . . n. Notice that (4.3a), (4.3b) and (4.3c) are consistent with (4.2a) and (4.2b). The Yuk variables satisfy (3.3a), (3.3b), (3.5) and (3.6') in the directed model because they satisfy constraints (2.3a). (2.3b), (2.5) and (2.6') in the undirected model. It remains to show that the directed variables Xi j satisfy (3.2), (3.4) and (3.7'). By definition the directed solution satisfies (3.4) for i < j. For the case i > j notice that (4.3c), (4.3b) and (2.4b) imply Xi j >- Mji q- Mi j - Mji = Mi j ( i = 2 . . . . . n, j = 1 . . . . . i - 1 ) , which shows that the directed solution also satisfies (3 .4 ) fo r i > j . As- sume, now, that some of the equality constraints

(3.2) are not satisfied by the directed solution. Then, (2.2), (4.2a) and (4.2b) imply that the left-hand side of some of the constraints (3.2) is greater than one while the left-hand side of some other constraints (3.2) is less than one. On the other hand, constraints (3.3b) and (3.4) (notice that we have already shown that the directed solution satisfies these constraints) imply that the left-hand side of every constraint (3.2) is greater or equal to one which leads to a contradic- tion. Finally, (4.2a), (4.2b) and (2.7') imply that the directed solution satisfies (3.7'). This shows that the reversed inclusion is also valid and the result follows. []

This result together with (4.2a), (4.2b) and the relationship between the costs of undirected and directed variables leads to v(UMCF L) = v(DMCF L) where v ( Q ) denotes the cost of the optimal solution of formulation Q.

Notice that the directed model involves only O(n 3) constraints while the undirected equivalent model involves O(n 4) constraints. This makes the directed model more suitable to use in lower bounding schemes for the HMST and as a consequence of that, the lower bounding schemes presented in this paper are directly derived from the directed formulation. However, the directed model cannot be used for "undirected" generalizations of the HMST problem. One ' such example considers the set of nodes V partitioned into two sets, O and D, and we want to find the minimal spanning tree subject to hop constraints between each node in O and each node in D. This generalization can be formulated as an undirected multicommodity flow model by introducing a different commodity for each pair of nodes i and j ( i ~ 0 and j E D ) and including the corresponding hop constraint. Notice that a directed model cannot be used in this case because the cost of an edge that carries flow in both directions will be counted twice. A further generalization consists of allowing some nodes to be in both sets, O and D. The extreme case O = D = V, consists of obtaining the minimal spanning tree with hop constraints between every pair of nodes is not greater than H. In this case it is sufficient to introduce a different commodity for each pair of nodes i and j ( j > i) and include the corresponding hop constraint.


5. Lifting the hop constraints for the directed model

By lifting the hop constraints we can derive sharper models for the HMST. For instance, the hop constraints can be lifted into:

i=0 j=l

k , v = 1 . . . . . n , k ~ v . (5.1)

The validity of (5.1) follows from the fact that the number of arcs in the path to node k cannot be equal to H if any arc ( k , v ) ( v = 1 . . . . . n; k ~ v ) is included in the solution. A few computational results taken from tests with n = 15 and which are presented in Section 9 show that for instances with H small the addition of such constraints produce significant improvements on the original lower bounds given by the DMCF L model. Notice that the validity of the proposed liftings depends strongly on the direction associated to the arcs. It is far from clear how to derive such class of liftings for the undirected model. This illustrates another advantage of using directed models versus undirected models. In the sequel, let L1DMCF denote the model DMCF with the set (5.1) instead of (3.5).

The disadvantage of the lifted model, is that we are replacing the original n hop inequalities (3.5) with a much larger set involving O(n 2) lifted hop inequalities. Additionally, many of the constraints (5.1) may not be violated by the optimal solution of DMCF L. Therefore, we also opted for considering a weaker lifted model which involves only a subset of the whole set (5.1). This subset involves the following n lifted constraints

~ ~ Y i j k < H - X k , v k i=Oj=l

k, v k -- 1 . . . . . n, (5.2)

where (k, v k) is the least cost arc leaving node k. From now on, L2DMCF refers to the model which includes the subset (5.2). The results given in Sec- tion 9 indicate that for many of the cases the bounds given by the weaker L2DMCF L are almost as good as the ones given by the stronger L1DMCF L.

6. An arc elimination test

The main disadvantage of using the DMCF L model (or the two stronger models defined in the last section) is that they still involve O(n 3) variables and O(n 3) constraints which lead to fairly large linear models even for small instances. In order to reduce the size of such models we present next a simple arc elimination test.

Arc Elimination Test. If Cij > C01 then no optimal solution does include the arc (i, j). If cij = Coj then there is an optimal solution which does not include arc (i, j) (i 4= 0).

Notice that if the optimal solution contains such an arc, then another feasible solution with the same cost (case c;j = c0j) or a better feasible solution (case cij > c0j) can be obtained by removing arc (i, j) and including the arc (0, j). Therefore, the arc (i, j ) can be eliminated from the problem and n - 1 constraints (3.4) can be eliminated from the directed model.

The results reported in Section 9 indicate that in some cases the arc elimination test leads to reasonable reductions on the size of the corresponding LP model. However, the reduced models are still too large for most cases, mainly when we want to solve instances with n ~ 20. This indicates that if we want to use the quality of the bounds given by the multicommodity flow model we ha~e to search for a!ter- native ways of computing that] bound. This can be accomplished by using lagrangean relaxation com- bined with subgradient optimization.

7. Lagrangean relaxations from DMCF

In this section we present two lagrangean relaxations which are directly derived from the DMCF formulation. To define the first one we follow Bal- akrishnan and Altinkemer (1992), attach non-negative multipliers l~ij k ( i = 0 . . . . . n , j , k = 1 . . . . . n, i 4= k) to the constraints (3.4), and dualize them in the usual lagrangean way. This leads to the following relaxed problem:

184 L. Gouveia / European Journal of Operational Research 95 (1996) 178-190

Relaxation DMCF1;~

m i n ~ ~ f l l j X i j + ~ ~ ~}~ijkYijk (3.1') i = 0 j = l i = 0 j = l k = l

subject to:

(3.2) , (3.3a), (3.3b), (3.5) , (3.6) and (3.7) ,

where

f l u = Cij-- ~ AUk i = 0 . . . . . n , j = 1 . . . . . n . k=l

Notice that for a given set of multipliers Azj. k, the relaxed problem DMCFla can be separated into two subproblems. The first one, RELxx, involves only the Xiy variables and can be easily solved by inspection. The other subproblem, RELrA, involves the Y~jk variables and can be further separated into n hop constrained shortest path problems, each one, RELrk ~, identified by the third index k ( k =

1 . . . . . n). To solve each subproblem RELrk x (k = 1 . . . . . n) let SP(0, k, q) be the cost of the shortest path between node 0 and node k and with exactly q arcs. This cost can be computed by the following Dynamic Programming recursion

SP(0, i, 1) = l~oi k i = 1 . . . . . n ,

SP(0, i, q) = min{SP(0, j , q - 1)

+ A~k I j = 1 . . . . . n, j # i}

i = 1 . . . . . n , q = 2 . . . . . H .

Then,

v(RELrkx) = min{SP(0, k , q ) [ q = 1 . . . . . H }

and

v(DMCFla) = v(RELxA ) + ~ v(RELrkx). k = l

An approximation of the optimal multipliers can be obtained by standard subgradient optimization (Held et al., 1974). A well-known result given in Geoffrion (1974) states that max[~lv(DMCFl~)_> v(DMCFL). Due to the presence of the hop constraints (3.5), each subproblem RELyk a does not satisfy the integrality property, which implies that the inequality above may be strict. In fact, the results reported in Section 9 indicate that for many cases we are able to improve on the bounds given by v(DMCFL).

Some preliminary results produced for some tests with n = 15 and 20 indicated that the convergence rate of the method is too slow. One way of speeding up the convergence of this method consists of adding to DMCF the directed subtour elimination constraints (3.8). In this manner, the subproblem in the Xij variables, RELxA, becomes a directed minimal spanning tree problem which can also be solved very efficiently by the method described in Fischetti and Toth (1993). According to the result given by Macu- lan (1986), the directed subtour elimination constraints (3.8) are already satisfied by DMCF L which implies that the theoretical best bound that can be obtained by this "augmented" procedure is not better than the theoretical best bound which could have been obtained by the original procedure. However, the convergence rate is much faster in the "augmented" procedure. To see this, notice that if the lagrangean multipliers are initialized to zero then the first lower bound in the sequence of iterations is equal to the cost of the corresponding directed minimal spanning tree. In many cases, this bound is much better than the one obtained by solving the corresponding inspection problem. In the sequel, this improved method is denoted by REL1.

To define the second lagrangean relaxation we attach multipliers 6jk (j , k = 1 . . . . . n) to the constraints (3.3a) and (3.3b), attach non-negative multipliers to k (k = 1 . . . . . n) to the constraints (3.5), and dualize them in the usual lagrangean way"

Relaxation DMCF2s,o,

min ~-~ ~cijXij-t- ~ ~-~ ~ otijkYijk i = 0 j = l i = 0 j = l k = l

j = l k = l

subject to

(3.2) , (3.4), (3.6) and (3.7),

where olij k "~- "l-O.) k -}" 6 i k - - ~ j k ( i , j , k = 1 . . . . . n , i :/= k ) and a o j k = + to k - 6jk ( j , k = 1 . . . . . n).

For a given value of the lagrangean multipliers to k and 6jk, the relaxed problem DMCF2~,,o can be solved using the following observation. Let Xo" be the optimal values of the X U variables in DMCF28,,o. Then, by using (3.4) and (3.6), the optimal values, Yo'k, of the Y~jk variables may be obtained by the


following rule: if aij ~ >_ 0 then _Yijk = 0 else _Yijk = X;j. This simply means that

OlijkYij k = min{0, aijk}Xij. (7.1)

Thus, constraints (3.4) and (3.6) can be dropped from DMCF2~,,o and the objective function (3.1") can be rewritten only with the Xij variables in the following way:

m i n ~ ~(Cij"[- ~min{O,o~ijg})Xij i = 0 j = l k = l

-t- ~ ~jj- ~ Hw k. (3.1") j = l k = l

The modified relaxed problem (3.1"), (3.2) and (3.7) involves only the Xij variables and can be easily solved by inspection. After the optimal Xo values have been obtained, we use (7.1) to obtain ~ e optimal values, Yijk, of the Yijk variables. Again, an approximation of the optimal multipliers can be obtained by standard subgradient optimization. Notice that the relaxed problem satisfies the integrality property. Therefore, we have maxts,,o]v(DMCF2~.,o) = v(DMCFL).

Following the suggestion given for the previous relaxation, we also add the directed subtour elimination constraints (3.8) to DMCF2~,,o. Then, the relaxed problem is also a directed minimal spanning tree problem. In the sequel, this method is denoted by REL2. Notice that the number of multipliers involved in REL2 is much smaller than the number of multipliers involved in REL1. Additionally, one directed minimal spanning tree has to be solved in each iteration of REL2 while one directed minimal spanning tree and n hop-constrained shortest-path problems have to be solved in each iteration of REL1. This means that REL2 is faster than REL1. On the other hand, the theoretical best bound we can hope to achieve with REL2 is weaker than the theoretical best bound associated to REL1.

8. A heuristic for the HMST

In this section we describe a slightly improved version of a heuristic given in Gouveia (1995). Briefly, this heuristic transforms a spanning tree into a hop-constrained spanning tree in the following

way: any node which is in position H * k + 1 (for any positive integer k) in the spanning tree solution is connected directly to the root. Clearly, for a node i satisfying this condition, the arc (p( i ) , i) where p(i) is the predecessor of node i in the original solution is removed from the solution. As an example consider H = 3 and the following solution with arcs (i, i + 1), i = 0 . . . . . 9. The arc exchanges performed by the heuristic connect nodes 4, 7 and 10 directly to the root node 0 and the original arcs (3, 4), (6,7) and (9, 10) are removed from the original solution.

Our improved heuristic uses slightly more complex are exchanges. To define these exchanges we start by defining an eligible node, i.e., a node candi- date for defining the new exchanges. A node is called eligible if it is located, as before, in a position H * k + 1 (for any positive integer k) and if its depth is less than H - 1 . The depth of a node is the maximum number of hops in any path starting from that same node. Considering again the same example, nodes 4, 7 and 10 satisfy the first condition for elegibility. Their corresponding depths are6, 3 and 0 respectively. Therefore, node 10 is the only eligible node. An eligible node i can be linked to any node which is (or is going to be) in any position L with L + depth(i) _< H. Clearly, we select the link of least cost for the exchange. Nodes which pass the first test for elegibility but fail the second test (nodes 4 and 7 in the example) are linked directly to the root as in the previous heuristic. In the example, nodes 4 and 7 are directly linked to the root. On the other hand, node 10 can be linked to any of the following nodes: 0, 1, 2, 4, 5, 7 and 8 (notice that after performing the exchanges associated to nodes 4 and 7, these two nodes will be connected to the root which implies that they will be in position one and nodes 5 and 8 will be in position two. Connecting node 10 to each of these nodes does not violate the required maximum number of hops for the solution. As pointed out before, we select the least cost arc for performing this more complex exchange.

This heuristic is embedded in the lower bounding schemes described in the last section, i.e., in each iteration of the subgradient optimization procedure the optimal relaxed directed spanning tree solution is transformed into a feasible solution for the HMST problem. One of the advantages of our heuristic is that we do not need to recalculate the position of


each node after each exchange is performed. Heuris- tics with more complex arc exchanges where one looks for the best savings feasible exchange could have also been used. However, such heuristics would be much more time consuming than the one proposed here. Additionally, the results presented in the next section indicate that with our simple heuristic we were able to produce the optimal solution for many of the instances tested.

9. Computational results

To assess the quality of our lower bounding schemes we have generated several groups of tests, euclidean and random. The costs for the euclidean tests were taken as the integer part of the euclidean distance between the coordinates of (n + 1) points in a square grid of dimension 100 by 100. The points were randomly distributed in this grid. For this class of tests we tested two root locations, one in the center of the grid of points (tests TC) and the other on the comer of the grid (tests TE). The costs for the random tests (tests TR) were generated in the inter- val [0, ~ 100]. For each group, TC, TE or TR we generated five complete graphs for three different

sizes, n = 20, 30 and 40. For each value of n we tried three different values for the hop parameter, H = 3, 4 and 5.

We also used one test of each group with n = 15 and H = 3, 4, 5 in order to compare the bounds of the LP models with the performance of the lagrangean relaxation schemes. These results are given in Table 1. The first three columns indicate the type of test, number of nodes and value of the hop parameter. The next columns indicate the lower bounds given either by the LP models or by the lagrangean relaxation schemes. We used CPLEX for solving the LP models. The bounds given for the lagrangean based schemes were rounded to the next integer. For the lagrangean based schemes we set a total of 250 iterations of the associated subgradient optimizatin procedure. The scalar used in the definition of the step size for updating the multipliers is usually set to a value between 0.0 and 2.0. In our procedure we found that the rate of convergence is significantly improved if higher scalars are used. We used 8.0 for initializing that value which was also halved whenever the bound does not improve in five consecutive iterations. Every fifty iterations, the scalar was set to a value equal to half its initial value. The results were obtained on a PC DX

Table 1 Bounds given by the LP models and the lagrangean relaxation schemes for the tests with n = ! 5 and H = 3,4, 5

Type n H MST WUMCF DMCF L 1DMCF L2DMCF REL 1 REL 11 R E L 2 REL22 OPT NEA

TC 15 3 298.46 304.5 306.33 306.0 311 310 301 304 313 (19) (37) (13) (13) (42) (48) (30) (31)

15 4 276 286.5 288.5 288.6 288.6 295 293 286 288 295 150 (< 1) (21) (28) (14) (19) (35) (55) (31) (31)

15 5 281.75 282 .33 283.2 283.2 289 290 279 281 293 (12) (5) (9) (12) (56) (63) (31) (31)

TE

TR

15 3 325.9 346.5 356.75 356.0 348 347 334 344 359 (611) (984) (1053) (763) (77) (88) (50) (51)

15 4 274 301.0 318.0 320.0 320.0 306 304 303 309 321 (< 1) (481) (733) (892) (818) (88) (100) (50) (51)

15 5 287.41 301.0 301.0 301.0 289 289 291 293 301 (431) (404) (677) (453) (99) (113) (51) (52)

15 3 178.50 182.25 186.88 185.46 193 193 180 182 193 (33) (42) (78) (89) (20) (41) (30) (30)

15 4 162 166.36 172.66 173 173 178 176 167 168 178 (< 1) (38) (82) (69) (53) (43) (53) (30) (30)

15 5 164 164 164 164 167 167 163 164 167 (29) (18) (89) (18) (26) (41) (30) (31)

56

153


486/66MHz. The lagrangean schemes were coded in Fortran. The corresponding CPU times are also given in seconds in the lines just below the corresponding bounds. We also included a column, MST, which gives the cost of the corresponding minimal spanning tree solution, a column, OPT, which gives the cost of the optimal solution and a column, NEA, which includes for each case the number of arcs eliminated by the test of Section 6. (Notice that a complete graph with n = 15 contains 225 arcs.)

These results seem to indicate that the bounds given by the LP relaxations of the multicommodity flow models are very sharp. Notice that L1DMCF L and L2DMCF L produce reasonable improvements on DMCF L when H = 3. Notice also that the bounds given by the two lifted models are almost the same which indicates that it may be reasonable to consider only the subset (5.2) of the whole set (5.1) corresponding to the lifted hop inequalities. Notice that for these instances, the bounds given by the directed model DMCF L are much better (in particular for the TE tests) than the bounds given by the unlifted undirected model WUMCF L. Due to the equivalence between the directed model DMCF L and the undirected model UMCF L these results also show what can be lost by replacing in the undirected model the bidirectional forcing constraints (2.4b) by the weaker unidirectional forcing constraints (2.4c).

Notice the discrepancy between the times for the TE tests and for the other tests. The reason for this is given by the arc elimination tests which performs much worse for the TE tests (see column NEA). This means that after reduction the LP models associated to the TE tests are in general much bigger than the models associated to the other tests.

For every T C and TR test REL1 gives better bounds than v(DMCF L) which confirms that with such a lagrangean scheme we are able in closing the duality gap. REL2 is much faster than RELI but, for these cases, the bounds given by REL2 are in general much worse than the bounds given by REL1. We also tested two other lagrangean relaxations, which use the lifted constraints (5.2). One such relaxation, REL11, is similar to REL1 but also dualizes the lifted constraints (5.2). The other, REL22, is similar to REL2, but dualizes constraints (5.2) instead of the original constraints (3.5). The results of Table 1 indicate that it is worth to consider the new method

REL22, in particular for the cases with H = 3. On the other hand, dualizing constraints (5.2) in the other scheme (RELll ) did not lead to significant improvements on the value of the bounds given by the original REL1. For the TE cases ( H = 4,5), the bounds given by REL1 and REL1 t are worse than the bounds given by REL2 and REL22. This contra- dicts the theoretical results stated before and might be explained by the ineffectiveness of the subgradient optimization procedure associated to REL1 and REL11 (notice that these two methods involves O(n 3) multipliers while the other two methods involve only O(n 2) multipliers). It is interesting to examine more closely the results for the TE tests. At first sight, such results indicate that it might be better to use the LP models instead of the alternative lagrangean based schemes. There are two reasons why such conclusions may be erroneous. In the first place we were not able to solve the LP models associated to most of the bigger tests (the only exception are the TC tests with n = 20 because the arc elimination test produces significant reductions on the size of the original LP models). For such cases we have to use the lagrangean based schemes. Secondly, the reported CPU times for these schemes are much smaller than the CPU times needed by CPLEX for solving the LP models. This means that we can increase the number of iterations for the lagrangean based schemes and improve the bounds given by the LP models. For instance, after 930 iterations by REL1 we obtained the optimal solution for the TE test with H = 3. Excluding the TC instance with H = 5, every other optimal value could have been obtained m a similar way. To obtain the optimal value for the TC instance with H = 5 we used the mixed integer version of the CPLEX code to solve the corresponding MCF model. By increasing the number of iterations of REL1 or RELl l we were not able in improving the lower bound value of 290.

In Table 2, we present the average gaps obtained by the four lagrangean relaxation schemes for the tests with n = 20, 30 and 40. The parameter values for the subgradient optimization procedures were the same as for the tests with n = 15. These tests were also performed in a PC DX486/66MHz. A column with label X indicates the average value of the ratio ( v ( U B ) - v ( X ) ) / v ( U B ) (where v(X) indicates the lower bound given by REL after X iterations of the


subgradient optimization procedure and UB indicates the best upper bound for the corresponding test) taken over the corresponding five tests. These upper bounds correspond to the best feasible solution obtained by the heuristic described in Section 8 and which was embedded in the lower bounding methods. We also considered a column (labelled by MST) which considers v(X) as given by the cost of the corresponding minimal spanning tree. Columns 1 to 3 describe the parameter values for the tests, columns 4 to 8 indicate the corresponding average gaps for each method, columns 9 to 13 give the corresponding average CPU times obtained after the 250 iterations. Column 14, NEA, indicates the average number of arcs eliminated for each group of tests. Notice

Table 2 Results given by

that the number of arcs in the original complete graph is equal to 400 in the n = 20 tests, 900 in the n = 30 tests and is equal to 1600 in the n = 40 tests.

Table 2 shows that the results for the tests with n = 20, 30 and 40 fol low the patterns shown in Table 1 for the smaller cases. For every TC and TR test with n = 20, the bounds given by REL1 and REL11 were almost optimal. Reasonable bounds are also obtained for the same class of tests with n = 30 and 40. In general the TE tests are more difficult than the TC tests. This might be explained by the fact that the arc elimination test performs much better for the TC tests. The random tests seem to be easier than the euclidean tests. The bounds given by REL1 and REL11 always dominate the bounds given by REL2

REL1, REL11, REL2 and REL22 for the set of complete graphs with n = 20, 30 and 40

Gaps CPU times (seconds)

Type n H MST REL1 REL11 REL2 REL22 REL1 REL11 REL2 REL22 NEA

TC 20 3 0.117 0.007 0.002 0.039 0.012 20 4 0.052 0.005 0.004 0.028 0.017 20 5 0.023 0.002 0.006 0.017 0.014

30 3 0.172 0.018 0.019 0.063 0.030 30 4 0.091 0.025 0.025 0.057 0.037 30 5 0.046 0.018 0.020 0.038 0.030

40 3 0.208 0.037 0.038 0.081 0.048 40 4 0.145 0.065 0.068 0.091 0.072 40 5 0.095 0.057 0.06t 0.076 0.067

E 20 3 0.227 0.037 0.046 0.073 0.040 20 4 0.139 0.051 0.056 0.062 0.042 20 5 0.087 0.044 0.050 0.051 0.044

30 3 0.306 0.112 0.121 0.128 0.085 30 4 0.224 0.140 0.149 0.131 0.107 30 5 0.162 0.118 0.122 0.107 0.090

40 3 0.340 0.179 0.196 0.136 0.119 40 4 0.270 0.214 0.224 0.169 0.147 40 5 0.212 0.189 0.192 0.162 0.144

R 20 3 0.287 0.008 0.014 0.147 0.098 20 4 0.148 0.021 0.035 0.098 0.083 20 5 0.064 0.017 0.020 0.044 0.042

30 3 0.289 0.020 0.043 0.158 0.102 30 4 0.143 0.035 0.051 0.105 0.084 30 5 0.074 0.032 0.039 0.067 0.059

40 3 0.361 0.035 0.050 0.190 0.121 40 4 0.189 0.049 0.064 0.145 0.119 40 5 0.109 0.053 0.065 0.102 0.088

90 100 62 63 104 117 63 64 278 98 110 52 53

284 318 193 196 335 375 195 199 629 385 433 196 199

562 705 286 291 743 829 285 291 1118 866 960 289 296

167 189 106 108 192 219 107 109 119 217 248 108 110

592 672 356 362 660 752 358 365 223 746 854 360 366

1275 1438 556 562 1465 1675 546 554 451 1670 1887 550 560

119 140 82 83 132 150 82 84 206 102 115 51 53

416 470 266 270 485 550 268 273 450 553 629 271 274

907 1032 310 314 1072 1212 312 317 838 1227 1391 313 318


and REL22 for the TC and TR tests. As expected, REL22 produced improvements on the bounds given by REL2. Notice that the REL22 method dualizes a set of constraints not involved in REL1. These re- suits indicate that the "non-integrality" property associated to the relaxed problem in REL1 seems to work better than adding the constraints (5.2) to the relaxation REL2. On the other hand, REL11 usually produces worse bounds than REL1 which indicates that adding constraints (5.2) to REL1 has not the same effect as using them in REL2. These results also indicate that the ineffectiveness of the subgradient optimization procedure associated to REL1 and REL11 becomes much worse for the larger tests. For such cases, the faster REL22 is clearly a better alternative.

As expected, the CPU times indicate that REL2 and REL22 are much faster than REL1 and REL11. Notice also that the CPU times of REL1 and REL11 depend strongly on the value of H which is explained by the number of constrained shortest path subproblems that have to be solved in each iteration of REL1 and RELl l . The excessively high CPU times associated to these two methods are explained by the fact that we are solving several hop constrained shortest path problems in each iteration of the subgradient optimization method and also by the fact that .we are dualizing O ( n 3) constraints. For instance, with respect to the TE tests with n = 40, more than 40000 constraints (3.4) are involved in the model. For a better comparison, no more than 1 second was needed for computing the MST bounds. The discrepancy between the CPU times associated to the tests with the root in the center and the tests with the root in the comer is explained by the number of arcs eliminated by the reduction test.

We have noticed that the bounds obtained after 250 iterations are in many cases still far from the theoretical best bound associated to the corresponding relaxation scheme. In order to have a better assessment of this theoretical bound and in order to evaluate the quality of the upper bounds given by the heuristic, we increased the number of iterations of the four methods to 1500. These results also indicate that REL22 converges much faster than the other two methods. The new gaps are indicated in Table 3 (for each test TE with n = 30 and 40 we used the best bound given by REL22; for the other cases we

Table 3 Best gaps and number of optimal solutions for each group of tests obtained with the methods discussed in this paper

TC tests TE tests TR tests

n H Gaps NOPT Gaps NOPT Gaps NOPT

20 3 0.000 5 0.008 3 0.000 5 4 0.001 4 0.012 1 0.001 4 5 0.000 5 0.014 2 0.001 4

30 3 0.001 4 0.062 0 0.000 5 4 0.004 2 0.062 0 0 .000 5 5 0.001 4 0.058 0 0.002 4

40 3 0.014 2 0.076 0 0.012 2 4 0.033 0 0.118 0 0.016 1 5 0.026 0 0.107 0 0.032 0

used the best bound given either by REL1 or REL11). For many of the smaller cases, the optimum was reached before the maximum limit of iterations. Col- umn NOPT indicates the number of optimal solutions obtained in each set of tests.

These results indicate that the solutions given by the heuristic are close to the optimum (in many cases, such solutions are optimal). The same results also indicate that from a theoretical point of view the proposed lower bounds are very sharp. In fact, the use of a branching scheme may not be needed if one is willing to spend the extra time in performing a large number of iterations for the bigger tests.

Finally, we pointed out that an interesting point of research consists of finding a different model whose LP value is precisely equivalent to the theoretical best bound associated to REL1 (as pointed out before this bound is better than the LP value of DMCF). Such a model might lead to alternative and much faster schemes for the HMST which produce bounds with the same quality of REL1. In particular, the proposed bounds for the larger TE tests might be significantly improved.

10. Conclusions

In this paper we discussed multicommodity flow formulations for a minimal spanning tree problem with hop constraints. We showed that a directed model is equivalent (in terms of the associated LPs) to a less compact undirected model. We derived several lagrangean relaxation based schemes and


presented computational results taken from a set of complete graphs with up to 40 nodes. The computational results indicate that for many cases the proposed bounds are sharp.

The HMST problem can also be generalized in several ways. Consider, for instance, the following weighted version of the hop inequalities constraints (3.5):

~_~ ~ wijYij~ < H k= l . . . . . n, (10.1) i=0 j = l

where wij is a weight (length) associated arc (i, j). For a given k, the "path-length" constraint (10.1) states that the weight of the path starting on the root node and ending at node k cannot be greater than H. From a reliability application point of view, consider the case where we associate a different reliability aij to each arc (i, j ) of the network. Such value may depend on the length of the corresponding link or the geographical location of the line associated to the same link. As before, one is interested in guaranteeing that each message sent from the computer site to each terminal has, at least, a certain probability/3 of reaching its destination. The reliability associated to each path P is now given by multiplying the indi- vidual reliabilities of the arcs in the path and the corresponding reliability constraint is given by

H Olij~-~/3" (i,j) e P

If one applies logarithms to each side of the above inequality we obtain

E aij > log( /3) (i,j)~ P

which can be modelled by the path-length constraints (10.1) if we associate binary variables to indicate whether an arc is in the solution and in addition if we multiply each side by - 1 and consider H = - l o g ( r ) and wij= --1og(t~i j ) for each arc (i,j). We can also associate weights qi to each node i (i = 1 . . . . . n). Then, constraints

~ qjYijk <--P k= l . . . . . n (10.2) i=0 j = l

state that the sum of the weights of the nodes included in each path starting from the root cannot be greater than p. These constraints might also

model a reliability application where one is con- cerned with the failure of nodes. Delay constraints with different delays associated to traversing the arcs (or traversing the nodes) can also be modeled by constraints (10.1) (or (10.2)).

Acknowledgements

The author thanks Geir Dalai from Norwegian Telecom for pointing out the reliability application mentioned in the introduction section. The author also thanks the referees for some helpful comments on the earlier version of this paper.

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Multicommodity flow models for spanning trees with hop constraints

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