Managing the Topological Expansion of Computer Networks in an Optimal Way This article describes a subgradient-based near-optimal heuristic algorithm designed for minimizing the search of links that need to be added to an existing telecommunications network to enhance the survivability and routability of the network.

By Debashis Saha and Amifava Mukherjee

Introduction n this article, a heuristic solution technique is presented for the problem of network expansion and the results of its compu- I tational experiments are presented. This

algorithm may form an integral part of a decision support system for network managers. It can be helpful for network designers too because the model presented here can also be extended to the general problem of designing a new network from scratch.

Debashis Saha received his BE in electronics and telecommunication engineeringfrom fadavpur University, India in 1986, and his MTech from the Indian Institute of Technology, Kharagpur, India in 1988. He was a research scholar in the Department of E 6 ECE, IIT until 1990, when he joined Jadavpur University as a Lecturer. His research interests include networks and distributed systems, parallel and dis- tributed processing.

Amitva Mukherjee received his BSc in mathemutics in 1979 and his MSc in applied mathematics in 1982, both from Jadavpur Univer- sity, India. He was a research scientist with the DOE Research Pro- ject in the Department of ETCE, Jadavpur University until 1989. He submitted his PhD thesis at Jadavpur University in the area of com- puter networks. His research interests include networks, parallel algorithms and distributed processing. His is currently with Price Waterhouse, India.

Topological expansion'-' of existing networks is a classic design problem in network management. The need for expansion may arise from several directions, such as perceived risk of link or node failure, inflation in demand for capacity from sub- scribers, a fall in operating charge, and an increase in estimated traffic due to growth in application. In topological expansion, the problem is to select appropriate links and nodes for addition to an existing network. A subproblem of the topological expansion problem is the link enhancement prob- lem3*4 where only extra links are added without adding any new node.

Today, the problem of network design is usually an evolutionary one in which the network is con- tinually expanded by adding new nodes and links. When a network is to be expanded, the goal should be to do so at a minimum cost. The man- agement objective can be identified as to generate as much profit as possible, considering the rev- enues generated vis-u-vis the corresponding costs of links. Usually a profit factor corresponding to each link is used to quantify the potential rev- enue generated.

Moreover, network survivability has received

206 IULY-AUGUST 1996

considerable attention over the last ten years mainly for the purpose of building fault-tolerant networks. This is also an important network man- agement problem because managerial consider- ations usually set a minimum budget target for enhancing network survivability. The easiest way to increase network fault tolerance is to add redundant links so that, in most cases, the surviv- ability consideration reduces to a combinatorial optimization problem involving a profit parameter vis-u-vis a cost parameter associated with added links5

Most previous papers on link enhancement have concentrated on finding algorithms based on either graph theoretic concepts, such as spanning trees and cut-sets,’ or a linear programming model2 for cost optimization. In reference 3, it has been shown that link enhancement is an NP-com- plete problem, and, hence, there is little likelihood of discovering a single algorithm that provides an optimal solution to all kinds of link enhancement problems. In trying to determine a near-optimal solution, several researchers have proposed heu- ristic approaches for solving only network surviv- ability problems ~eparately.~,~

opological expansion of existing T networks is a classic design problem in network mangement.

There has been little study on the relative merits of available heuristic algorithms, except what could be found in the work of Yang3s4 To our knowledge, there has been no minimization algor- ithm based on the subgradient technique7. Since Lagrangean relaxation combined with subgradient optimization forms a very good technique for non- linear combinatorial optimization problems5, we have made an attempt to apply the technique to the link enhancement problem. The justification of the proposed heuristics rests on an intuitive notion which is supported by simulation results. Our analysis is based on how well our heuristics mini- mizes cost for a sample set of randomly generated networks. Thus, we avoid the bias associated with a specifically chosen network.

Network Model

Let us consider a computer network represented by a graph G(V,E), where V represents the set of vertices (or nodes) and E the set of links (or edges). In general, some nodes are directly connected to each other via communication links, whereas the rest of the nodes are indirectly connected to each other via intermediate nodes. In order to expand the network, it is sometimes desirable to convert some of the indirectly connected nodes to directly connected ones by adding more communication links to the network. Such link enhancements increase network routability and servivability. For an existing computer network, it is an important mangerial question to find the optimal link enhancement for a given investment while main- taining maximum network survivability and opti- mum routability. However, we are not considering survivability and routability in this article for the sake of simplicity.

Let ci j represent the cost to establish the link between node i and node j (i,j E V), and p,, the per- formance contribution (profit) when the link between nodes i and j is established. The profit pi j is a function of network and/or survivability measure^.^,^ So the link enhancement problem can be stated formally as follows.

Given:

Find:

such that

a budget ( B )

the best link enhancement strategy ( L )

(1) pji is maximized, and id

( 2 ) 2 cij < B ‘ 4

The complexity of the problem has long been well known.’ An exhaustive algorithm to solve it requires exponential time. As a result, consider- able effort has been devoted to develop heuristic methods. Such a heuristic normally proceeds in two basic steps:

(1) Select a link and ( 2 ) Update the budget.

The link can be selected in a random fashion (greedy algorithm), or the most profitable links can be selected first, or the most profitable links

INTERNATIONAL IOURNAL OF NETWORK MANAGEMENT 207

are chosen first and, at the same time, an attempt is made to increase the cost as slowly as possible. A brief description of the algorithms and the sub- sequent motivation behind the present work is dis- cussed in the next section.

An Overview of Minimization Algorithms

In this section, we first describe four previously proposed synthesis methods. They have been con- sidered in detail in references 3 and 4. In these approaches, a link is first chosen. Next, its cost is subtracted from the total budget. The process is repeated until no more links can be accommo- dated. The heuristics under consideration differ in link selection strategy mainly:

0 Heuristic 1: In this method, the link is chosen randomly with all choices being equally likely. That is, at each stage of the process, one link is chosen randomly from among all candidate links, and its cost is subtracted from the budget. The process is repeated on the remaining links. Since no particular character- istics of the link are used to determine the choice of the link, this simple heuristic pro- vides a basis of comparison for the other heu- ristics which use some characteristics of the links to limit, in varying degrees, the choices of links.

0 Heuristic 2: In this heuristic, the link is chosen as the one which has the minimum cost. If there are two or more such links, the one which has a larger profit is chosen. If profit is also identical, one of them is chosen ran- domly-this is how ties among links having the same cost factor are broken.

0 Heuristic 3: In this approach, each link is first assigned a weight which is calculated as the ratio of cost to profit. The link with the small- est weight is chosen. If there are two or more links having an identical weight, the one with. the less cost is chosen. The basic idea of this heuristic is to cover the least-cost, most- profitable link first.

0 Heuristic 4: This heuristic is similar to Heuris- tic 2. In this approach, a factor is calculated for each link. This factor is inversely pro- portional to the link weight described in Heu-

ristic 3. The link with the least cost is chosen. If there is a tie, the one with the highest factor is chosen.

We have implemented all the heuristics in C on a HI' 9000 minicomputer and tried them on several networks. No single heuristic has been found to be equally effective for all kinds of networks. The algorithms display a diversity in realizations. That is, no single algorithm has been found to be con- sistently better than others over all networks. There are classes of networks where one heuristic is better than others. This motivates us to try another well-known optimization technique, namely subgradient optimization, to this problem. Since this problem can be easily formulated as a non-linear combinatorial optimization problem, similar to the one described in reference 5, we can adapt the algorithm of reference 5 to this probIem with little modification. So the next heuristic is proposed:

0 Heuristic 5: This is a subgradient heuristic sol- ution procedure, originally designed for opti- mization problems and developed in conjunc- tion with the Lagrangean relaxation te~hnique.~ This procedure attempts to gener- ate a feasible solution for the optimization problem, after every iteration of the subgradi- ent optimization algorithm.s The best solution is retained when the subgradient algorithm terminates. The subgradient optimization algorithm derives lower bounds on the primal objective value. Heuristic 5 is discussed in detail in the following sections.

Problem Definition Let xij represent a decision variable defined as

follows:

xu = 1 if the link (i-1) is to be connected

= 0 otherwise

Let us suppose that the input data set consists of tuples of the form (i,j,c,,,pJ, where i and j are the node numbers of the network, c,,, the cost to estab- lish the link (i-j) between nodes i and j , and p,], the profit or the contribution of this intercon- nectivity (or link enchancement).

The problem is defined as a maximization prob-

208 JULY-AUGUST 1996

lem for a given budget. The outline of the prob- lems is given as:

Max: z, = C&P,P I /

where a is a computational parameter (a>1>.5 Subject to:

cc XeC, < 6

o s x , < 1 / I

The objective function represents the profit gener- ated of assigning the links to the nodes. Constraint (1) ensures that the budget estimation must be sat- isfied for link enhancements.

Heuristic Solution Procedure The complete procedure consists of two subpro-

cedures which are described in sequence here.

- Lagrangean Relaxation of the Problem -

The model described above is a non-linear com- binatorial optimization problem. Since the decision analog of this model belongs to the NP- complete class of problems", Lagrangean relax- ation is used to develop a heuristic solution to this problem. The Lagrangean relaxation scheme has been earlier successfully applied to many combi- natorial design problem^.^,'^ The Lagrangean relaxation (G) of the problem is formed by multi- plying constraint equation (1) by vectors of the Lagrangean multiplier A, and then by adding that to the objective function:

subject to constraint equation (2). A solution of Lagrangean relaxation is then

obtained as follows. Differentiating G partially with respect to xii and setting aG/ax, equal to zero, we obtain:

aGlax, = appq-1 + A,/c, = o

or

If a! = 2, then xj j = [-Ai,cii/2p;].

-Subgradient heuristic - Let us suppose that A* be an optimal solution of

Lagrangean relaxation. The subgradient optimiz- ation algorithm is used to derive lower bounds on the optimal primal objective value using G. In the subgradient optimization procedure the gradient method is adapted by replacing the gradients with subgradients.s If an initial multiplier vector A, is given, a sequence of multipliers is generated using the following expression:

where xz is obtained from an optimal solution to Lagrangean relaxation G and t,, a positive scalar, is stepsize which is given as follows:s

where 6, is a scalar satisfying 0 4 6, Initially, this scalar is set equal to 2, and it is

then halved when the lower bound does not improve in a given number of consecutive iter- ations. The subgradient algorithm is terminated, if either the gap between the upper bound (the value of G) and the best primal feasible solution value are within a given specified limit, or after 300 iter- ations, whichever is earlier. The details of the algorithms is given in Reference 6.

2.

Computational Results We have applied our heuristics to the problem

of Table 1 taken from reference 3. For a budget of 7000 units, the links selected are L, = l3,7,8,171 whose costs sum to 6245 units. The total profit obtained is 16,523 units. The selected links are marked by asterisks in Table 1.

Next, we have applied the heuristic to the prob- lem of Table 2 which is again taken from reference 3. For a budget of 11,500 units, the links selected are L, = (7,1,12,5,S,4,11,13,1S,17,5,21 whose costs are 11,434 units. The total profit obtained is 37,862

INTERNATIONAL IOURNAL OF NETWORK MANAGEMENT 209

Link number cost Profit

1 2 3* 4 5 6 7* 8* 9

10 11 12 13 14 15 16 17* 18 19 20

1833 1754 1246 1529 2034 2568 1508 1608 1691 21 12 1 840 1960 21 84 2549 2254 2289 1883 1682 171 1 1578

41 40 3506 3819 2310 3370 5276 3859 4477 3269 3807 366 1 3560 4440 2899 3643 4224 4368 1922 3844 3484

Table 1. (n = 20, B = 7000)

Link number cost Profit

1* 2* 3 4* 5* 6 7* 8* 9

10 11* 12* 13* 14 15* 16 17* 18* 19 20 21 22 23

796 1225 1066 847

1003 1057 693 823

1498 1042 89 1 91 6 957 787

1031 1067 1082 1187 1107 964

1012 1196 80 1

3191 2963 291 6 3104 2947 9026 3314 3074 31 07 2824 3140 327 1 31 32 2746 3304 2870 31 27 3295 2900 281 2 2868 281 8 2762

units. The selected links are marked by asterisks in Table 2. The same problem has also been solved by Yang3f4 with the help of his CRCS method. The solution set contains 12 links, namely L3 = {18,17,15,13,12,11,8,7,5,4,3,1 I, whose total profit contribution is 37,815. The exact optimal solution set for the problem of Table 2 is, however, given by L4 = {18,17,15,13,12,11,8,7,5,4,2,11 whose total profit contribution is 37,862 units. Though this set differs from L3 by one link (namely, link 2 instead of link 3), L4 is identical to Lz. This proves the efficiency of our technique over Yang's method for this example. Moreover, in this example problem, the solution set by the subgradient heuristic is obtained in four iterations only, as shown in Table 3. In each iteration, the link set L' is improved as indicated by the total profit P'.

Conclusion We have studied link enhancement/design

problems, and a mathematical model is presented to estimate demand and to decide enhancement strategy in an expanding network for a profit-max- imizing design. The model captures link profit in the objective function. A heuristic solution tech- nique has also been presented for the problem of network enhancement based on the profit model.

As discussed earlier, the basic aim of the net- work enhancement problem is to choose which links to add to an existing network in order to maximize profit. The problem is an evolutionary one in which the network is continually expanded over time. To reflect this in our model, we use a capacity budget constraint on the overall capacity of new link installations during the current plan- ning period.

As more and more nodes and links are added over time, the network may evolve to a general mesh structure. In some cases, the topology may change so much that the protocol suite may need to be modified. This must also be kept in mind while modifying a network on a large scale.

Table 2. (n = 23, B = 11500)

210 JULY-AUGUST 1996

Iteration i Link Set L' Total cost C Total profit P

1 2 3 4

{7,1r12,8,4,11,15,13,14,23,18,17} 10.81 1 {7,1,12,8,4,11,15,13,14,5,18,17} 10,013 {7,1.12,8,4,11,15,13,5,3,18,17} 1 1,292 (7,1,12,8,4,11,15,13,5,2,18,17} 1 1,451

37,661 37,846 37,815 37,862

n some cases, the topology may change I so much that the protocol suite may need t o be modified.

References 1. T. H. Wu, R. Cardwell and M. Boyden, A multi-per-

iod architectural selection and optimum capacity allocation for future SONET inter office networks, ORSA/TIMS Joint National Meeting, PA, October 1990.

2. L. J. Leblane and S. Narasimhan, Topological expan- sion of metropolitan area networks, CN & ISDN Sys- tems, 26, 1235-48, 1994.

3. C. Yang, Link enhancement using constrained range and reduced candidate set searches, Computer Com- munication, 15, No. 9, September, 573-80, 1992.

4. C. Yang and R. Lin, Budget management of network capacity planning by searching constrained range and dominant set, I E E E Journal Select Areas Commun.

5. D. Saha, A. Mukherjee and S. K. Dutta, Capacity assignment in computer communication networks with unreliable links, Computer Communication 17, No. 12, December 1994.

12, NO. 6, 301-8, 1994.

6. D. Saha and A. Mukherjee, On the topological expansion of computer networks, Technical report, Dept of Computer Science and Engineering, Jadavpur University, March 1995.

7. M. L. Fisher, The Laeraneean relaxation method for " " solving integer programming problems, Manage. Sci. 27, 1-18, 1981.

8. H. Held, P. Wolfe, and H. P. Crowder, Validation of subgradient optimization, Math. Program. 5, 62- 8, 1974.

9. M. L. Fisher, An applications oriented guide to Lag- rangean relaxation, Interfaces, 15, 10-21, 1985.

10. M. S. Bazaraa, and J. J. Goode, A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangean duality, Euro. J . Operaf. Res.

11. M. R. Garey and D. S. Johnson, Computers and Intrac- tability: A guide to the theory of NP-completeness, Free- man, San Francisco, 1979.

3, 322-8, 1979.

If you wish to order reprints for this or any other articles in the International Journal of Network Management, please see the Special Reprint instructions inside the front cover.

CCC 105.5-7148/96/040206-06 0 1996 by John Wiley & Sons, Ltd

INTERNATIONAL IOURNAL OF NETWORK MANAGEMENT 211