A Lagrangean approach to the degree-constrained minimum spanning tree problem

European Journal of Operational Research 39 (1989) 325-331 North-Holland

325

Theory and Methodology

A Lagrangean approach to the degreeconstrained minimum spanning tree problem A. V O L G E N A N T

Department of Operations Research, Institute of Actuarial Sciences & Econometrics, Faculty of Economics and Econometrics, University of Amsterdam, Jodenbreestraat 23, 1011 NH Amsterdam, Netherlands

Abstract: A known branch and bound algorithm for the degree-constrained minimum spanning tree

problem is adapted for the use of Lagrangean multipliers. They improve the bounds from which the edge elimination analysis in the algorithm benefits in particular. Computational results are given for problems up to 150 nodes both random drawn coordinate problems as well drawn table problems. These results are much better than other results reported in literature and they show the Lagrangean approach to be more effective when the problems are more difficult to solve. Keywords: Degree-constrained trees, Lagrange multipliers, branch and bound

1. Introduction

Since Kruskal (1956) published his greedy algorithm for the Minimum Spanning Tree (MST) problem, many applications of this algorithm have been found: Held and Karp (1970, 1971) for instance, applied the MST as a lower bound for the Traveling Salesman Problem. Variations of the MST problem occur in the field of communication and computer networks. One of them is the Degree-Constrained Minimum Spanning Tree (for short DCMST) Problem, stated as: Given an undirected, complete graph G = (V, E ) with weights (costs or length or time) c u associated with each edge (i, j ) ~ E, ci~ = or, find a spanning tree of minimum total weight, such that the degree di at each node i ~ V is at most a given value b~. Received September 1985; revised December 1987

Narula and Ho (1980) mention an example in designing computer networks when connecting terminals with a minimum amount of wire; sometimes the size of the terminals is such that the number of wires incident to a terminal i cannot exceed a number b i and then the minimum total length of the wire is given by the optimum of the D C M S T problem. Gavish (1982) gives many references for this type of problems. Furthermore the D C M S T can be a subproblem for solving other problems, e.g., the capacitated MST problem, as done by Gavish (1985) in centralized network design. Garey and Johnson (1979) showed the DCMST problem to be NP-hard by reducing it to an equivalent symmetric Traveling Salesman Problem, and so it is unlikely that a polynomially bounded algorithm exists for solving the general D C M S T problem. For the special case with a degree constraint for only one node, Gabow and Tarjan (1984) developed an efficient algorithm with time complexity of O ( E + n log n). Yama-

0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

A. Volgenant / Degree-constrainedminimum spanning treeproblem

326

moto (1978) suggested solving the DCMST problem by finding a common basis of two matroids. Narula and Ho (1980) gave two heuristic procedures and a branch and bound algorithm to construct a DCMST. Gavish (1982)presented successful computational values for Lagrangean based lower bounds, using a subgradient optimization procedure to compute the multipliers, while here a dual ascent procedure is applied in a complete branch and bound procedure. Savelsbergh and Volgenant (1985) described a branch and bound algorithm based on edge elimination used for the Traveling Salesman Problem. Our purpose is to describe an algorithm with a better performance, at least in the case of large random table problems, based on Lagrange multipliers as well as on edge exchanges. In the following section the DCMST problem is described and in Section 3 the determination of the Lagrange multipliers. Section 4 overviews the branch and bound procedure; Section 5 gives the computational results in comparison with known results. The last section gives some conclusions.

The DCMST problem can be formulated as a linear 0-1 integer programming problem. Let n =

IV I and if an edge (i, j ) ~ E is included in a tree, otherwise.

Then the problem is: (PC)

min

i

n cijxij Y'. i=l j=l

(1)

subject to n

bi,

i= l,...,n,

(2)

k Xij ~ 1,

i=l,...,n,

(3)

for all N c V,

(4)

i,j=l

(5)

E Xij ~ j=l

j=l

x~j= INI-1

i,jEN Xij = 0

or

1,

. . . . . n.

Because of symmetry, cij = cji.

n

(PC,~)

(Cij+'lTi)Xij--~rtib

rain ~'. ~ i=1

j=l

i

(6)

i=1

subject to 7r/>~ 0,

i = 1 . . . . . n,

(7)

and (3), (4) and (5). According to Geoffrion's Theorem 1 (1974) solution X of PC is optimal if X satisfies, for a given ~r, three conditions: (i)

X is optimal in PC,r, n

(ii)

Y'~ x i j <~ b,,

i = 1 ..... n,

j=l

(iii)

q'fi

Xij

-- b i

= O,

i = 1 . . . . . n.

Thus PC,~ is a lower bound for the DCMST problem, and the best bound of this type is PC* = max = > 0PC~.

2. The problem and its Lagrangean

x i j = {1°

Everett (1966) suggested weighting constraints by multipliers and adding them to the objective function. If we apply this to problem PC we obtain the Lagrangean:

3. Determining the Lagrange multipliers The quality of the bounds is important in a branch and bound algorithm; so the Lagrangean approach can be valuable in improving the results, especially of the more difficult large random table problems. However, finding a good set of multipliers is not simple and the potential usefulness of a Lagrangean relaxation is largely determined by how near its optimal value is to that of the original problem; if X only satisfies (i) and (ii), but not (iii) then we have a so-called duality gap. The determination of a (good) set of multipliers will be called the ascent. In an ascent the multiplier values % are initially set to zero, but are chosen positive for i with d i > bi. In order to satisfy condition (iii), we will allow a decrease of these values when d~ < b~ for nodes with 7ri > 0. The ~r-values are changed as suggested by Volgenant and Jonker (1983): ~i m+l

=

max(0, ~rim + 0.6 t m ( d m - bi) +0.4tm(dm-l-bi)}

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A. Volgenant / Degree-constrained minimum spanning tree problem

with step length t m a positive scalar decreasing to zero during the ascent according to a series with constant second order difference:

Table 1 Computational results for a random table problem with size n = 50 and constraint 3 Quantity

t m+ l

--

2 • t m + t " - 1 = constant.

The characteristics of the ascent are: - the m a x i m u m number of steps is fixed on N=3.6 with 6 the number of times the constraint is violated in the first MST; this choice follows from the expectation that the value of N a measure is for how difficult a problem will be; -we set d ° = bi, for all i ~ V and t 1 = N * 7"/30, with T the difference of the first upper and the first lower bound; - as long as the second value in the iteration is smaller than the first one, the step length is reduced with 10%, to avoid the use of too large steps; - for trees with only a few nodes violating the degree constraints the branching part can in general faster solve the remaining problem than the bounding part, so the iteration is terminated prematurely when a lower bound solution is found within 1% of the upper bound and in which n

E (dj-b)~
for all i with d, > be,

and d~ ~< b~ for all other nodes. Some limited computational experience gave the right hand side as a good choice. The effect of an ascent on the quality of the lower bound is substantial: for the example problem in Section 4, Table 1, the bound improves from 8.1% to 0.6% under the optimum. If the MST that is a solution of PC~, satisfies conditions (i), (ii) and (iii) during or at the end of the ascent the problem has been solved; otherwise a branch and bound procedure is applied that will be outlined in the next section.

4. T h e b r a n c h a n d b o u n d p r o c e d u r e

In case the problem was not solved during the ascent a branch and bound procedure has been applied together with an edge analysis procedure, both as described by Savelsbergh and Volgenant (1985). For the convenience of the reader we outline these procedures together with some adaptions due to the use of the Lagrange multipliers.

Without multipliers

With multipliers

Best lower bound (% under optimum 1329) 1221 (8.1%) 1321 (0.6%) Best upper bound (% above optimum 1329) 1332 (0.2%) 1356 (2.0%) Number of forbidden edges (% of total 1225) 1030 (84.1%) 1137(92.8%) Number of required edges (% of total 50) 0 (0%) 13 (26%) Number of subsets (trees) 104 (126) 5 (14) CPU time in seconds 37.2 3.4

They improve the lower bounds and as a consequence the effect of the edge analysis procedure as will be illustrated by interesting figures in Table 1 for some average examples. The ascent has only be used in the general problem and not in the subproblems generated in the branching process; the ascent appeared to be too time consuming for the subproblems and did not sufficiently improve the quality of the lower bounds. Besides the better lower bound produced by this ascent it is also an advantage to have more trees as a basis to compute upper bounds according to the heuristics A H and C H as introduced by Savelsbergh and Volgenant (1985). This advantage is so substantial that the much time consuming heuristic of Narula and H o (1980) has been omitted from the algorithm. In each subset of the branching an edge analysis procedure has been applied if a different lower bound tree or a better upper bound is found. This procedure forbids (requires) an edge if its inclusion (exclusion) increases the lower bound above the current upper bound. All the logical consequences of forbidden or requiring an edge are drawn immediately. To start the branching a node i is chosen for which d i - b i is positive and minimal. If the value of i is not unique, we take the one with a maxim u m number of required edges, hoping to keep the number of branches in our search tree small, at least at the beginning of the branching. Branching then takes place by forbidding a n d / o r requiring edges connected to this node.

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A. Volgenant // Degree-constrained minimum spanning tree problem

Table 2 Average number of required and forbidden edges in % of the total number of edges for random table problems Constraint value b

Savelsbergh and Volgenant

With Lagrangemultipliers

n

Required edges (%)

Forbidden edges (%)

Required edges (%)

Forbidden edges (%)

30 50 70 70 70 100 100 100 100 a 150 a

3 3 3 4 5 3 4 5 3 3

6.7 2.6 0.2 33.3 67.0 0 22.2 71.2

75.8 81.1 77.9 87.7 90.0 86.7 97.0 99.3 -

27.3 53.2 21.6 68.3 80.1 25.6 62.0 97.1 60.6 21.5

76.2 94.6 96.4 96.4 98.1 93.1 96.4 99.2 98.5 97.5

Size

-

-

- Results not available. a With fictitious upper bound.

If condition (ii) is satisfied, but (iii) not, we still have to branch. Therefore, if we do not end with a tree with di >/b i for all i with % > 0 or if a tree, not satisfying this condition, is generated in a subset, we adapt the multiplier set by successively decreasing multipliers ~r~ with d~ < b~ until d~ >i bg or ~r~= 0. We chose for linear decreasing in (at most) three steps to zero as it is a simple heuristic rule which leads to satisfying results. In Narula and Ho (1980) for a random table and a Euclidean problem all heuristic solutions have been given. Our solution for the Euclidean problem is not significantly different, but due to the Lagrange multipliers the quality of the lower bound and so the impact of the edge analysis procedure for the random table problem improved drastically--see Table 1 for an example also given by Savelsberg and Volgenant (1985). Table 2 shows the average effect of a better lower bound on the analysis, again compared to the results of Savelsbergh and Volgenant (1985). The new results are given for the edge exchange analysis after the ascent. As the upper bound values are obtained in a different way for the two methods (see above) the results in Table 2 are average indications for the quality of the method and the level of difficulty of the considered problems. The comparison of the results for different constraint values shows the expected result that problems become easier with larger values of the constraint. For the Lagrange version and the large problems with size 100 and 150 a fictitious upper bound value has been used

equal to L * (1 + 8/1000) with L the lower bound value and 8 the number of times the constraint is violated in the best lower bound tree.

5. Computational

results

The results presented first in this section are based on the same problems, ten for each problem size, as were solved in Narula and Ho (1980) and in Savelsbergh and Volgenant (1985), which gives a good comparison. Due to the Lagrange multipliers (and the edge analysis) we were able to solve the more difficult table problems with n = 100 on a personal computer in a shorter time (on the average) than Narula and Ho (1980) needed for the easier Euclidean problems of the same size on a main frame computer (CDC Cyber 173). For convenience we only considered problems with degree constraints b i = b (i = 1. . . . . n), but the program can be easily adapated to problems where different values of b i are given. The algorithm in the given form is not suited for problems with b = 2. An additional ascent in the subsets would be necessary. However, the solutions for such problems can be easily obtained from the TSP solution by deleting the largest edge. For the TSP problem there exist special algorithms, e.g., Held and Karp (1970, 1971), and Volgenant and Jonker (1983). None of the Euclidean problems with dimension 2 and size n = 50, 100 and 150 needed the ascent and the results of Savelsbergh and Volge-

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A. Volgenant / Degree-constrained minimum spanning tree problem

Table 5 Averages for different values of the constraint and the dimension; the size of the problems is n = 70 (starred results are uniformly better per problem) Type of problem

Constraint

Number of trees

Dimension 3

3 4

12.0 1.2

6.7 1.5

7.5 1.3

3.6 1.0

Dimension 4

3 4

39.1 1.5

15.9 2.5

18.8 1.7

7.9 1.6

Dimension 5

3 4

325.0 1.4

30.4 * 2.6

131.2 2.4

14.1 * 1.9

Random table

3 4 5

> 2662 38.2 1.8

27.9 * 8.4 2.8

> 116 a 17.2 1.9

10.6 * 3.7 1.4

S&V

CPU seconds Lagrange

S&V

Lagrange

a CDC Cyber 175 seconds; all the other times obtained on a personal computer.

edge e l i m i n a t i o n p r o c e d u r e r e d u c e d the b r a n c h i n g part considerably. F i n a l l y we tested the following types o f data. I n a k - d i m e n s i o n a l cube n p o i n t s were g e n e r a t e d , using the d i s t a n c e s b e t w e e n these p o i n t s as the cost entries, with k = 3, 4, 5 (the case k = 2 corres p o n d s to the b e f o r e m e n t i o n e d E u c l i d e a n p r o b lems). F o r each value o f k ten p r o b l e m s were g e n e r a t e d a n d T a b l e 5 gives average results for b=3 a n d 4 as well as for the r a n d o m t a b l e p r o b l e m s ( r e p e a t e d p a r t l y f r o m T a b l e 4 for convenience). Results for b = 5 are o n l y given w h e n there are differences for the two a p p r o a c h e s : for d i m e n s i o n 3, 4 a n d 5 all first trees o b e y a l r e a d y the degree c o n s t r a i n t b = 5. A s was to b e expected, the p r o b l e m s are m o r e difficult with increasing value of the d i m e n s i o n k a n d with d e c r e a s i n g c o n s t r a i n t value b, while the L a g r a n g e a n a p p r o a c h b e c o m e s m o r e effective.

F r o m the results it is o b v i o u s that the L a g r a n g e a n a p p r o a c h is m o r e effective when the p r o b l e m s are m o r e difficult to solve. A l t h o u g h results are m i s s i n g for p r o b l e m s w i t h m o r e t h a n 150 nodes, we t h i n k t h a t the d e s c r i b e d a l g o r i t h m is a b l e to solve larger p r o b l e m s in r e a s o n a b l e times. T h e given c o m p u t a t i o n a l results i l l u s t r a t e (again) the success o f the L a g r a n g e a n a p p r o a c h in combinatorial optimization.

Acknowledgement T h e a u t h o r is grateful for the p o s s i b i l i t y to use s o m e c o m p u t e r p r o c e d u r e s w r i t t e n b y M. Savelsb e r g h a n d for the a s s i s t a n c e o f J . H . M . G o m m e r s a n d P. B e r k h o u t in p r e p a r i n g the c o m p u t a t i o n a l results.

References 6. Conclusion L a g r a n g e multipliers are used to c o m p u t e b e t ter lower b o u n d s , t h a t is: m u c h closer to the o p t i m u m . T h e trees in the ascent are e x p l o i t e d to generate u p p e r b o u n d values. T o g e t h e r with the edge exchange analysis t h e y f o r m the basis of a fast a l g o r i t h m for the D C M S T p r o b l e m . E s p e cially the t e r m i n a t i o n of the i t e r a t i o n p r o c e s s w h e n a r e a s o n a b l e g o o d s o l u t i o n is found, w o r k s t i m e saving.

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A. Volgenant / Degree-constrained minimum spanning tree problem

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