graduate school of information, production and systems, waseda university 7. minimum spanning tree...

Graduate School of Information, Production and Systems, Waseda University

7. Minimum Spanning Tree Problem

Soft Computing Lab. WASEDA UNIVERSITY , IPS 2


1. Multicriteria Minimum Spanning Tree (mc-MST)1.1 Basic Concept of mc-MST1.2 Genetic Algorithms Approach1.3 GA procedure for mc-MST1.4 Numerical Experiments

2. Degree-constrained Minimum Spanning Tree (dc-MST) 2.1 Basic Concept of dc-MST2.2 Genetic Algorithms Approach2.3 GA procedure for dc-MST2.4 Numerical Experiments

3. Degree-based Permutation GA for dc-MST3.1 Concept on Degree-based Permutation GA3.2 Genetic Algorithms Approach3.3 Degree-based Permutation GA for dc-MST 3.4 Numerical Experiments

4. Leaf-constrained Minimum Spanning Tree 4.1 Basic Concept of lc-MST4.2 Genetic Algorithms Approach4.3 GA procedure for lc-MST4.4 Numerical Experiments



The Minimum Spanning Tree (MST) problem was first formulated by Boruvka in 1926 when he developed a solution to finding the most economical layout of a power-line network. Graham, R. & P. Hell: On the history of the minimum spannin

g tree problem, Annals of the History of Computing, vol. 7, pp.43-57, 1985.

Since then the minimum spanning tree formulation has been widely applied to many combinatorial optimization problems:

Transportation problems Telecommunication network design Distribution systems etc.

Kershenbaum, A.: Telecommunications Network Design Algorithms, McGrawHill, New York, 1993.

Introduction



Minimum Spanning Tree (MST) problem is one of the traditional optimization problems.

Given a finite connected graph, the problem is to find a least-weight subgraph connecting all vertices.

(i, j) wij

(1, 2) 2

(1, 6) 5

(1, 8) 3

(2, 3) 4

(2, 8) 6

(3, 4) 2

(3, 9) 2

(4, 5) 1

(4, 9) 8

(5, 6) 6

(5, 7) 2

(6, 7) 7

(7, 8) 1

(7, 9) 4

(8, 9) 3

Data table of example network

Basic Concept of Minimum Spanning Tree Problem

1 2

6

8

37 9

5 4

2

5

3 6

74

1 3

2

4

6

1

2 82

i jwij

Tavakoly., B.: Gene Expression Data Clustering With Minimum Spanning Tree, Department of Information Systems and Computing, Brunel University, May 2003.

Table for non-directed graphFig. 7.1 Example of network model



Notations Indices

i, j : the index of node, i, j =1, 2, …, n

Parameters

n: the number of nodes in the network

V: the finite set of nodes (vertices) representing terminals

S : the subset of nodes

wij : the weight of connecting node i to node j, i.e., the weight of

link (i,j); the weight matrix (wij) is symmetric.

Decision Variables

xij : the 0,1 decision variable; 1, if link (i,j) is selected, and 0, otherwise.


7. Minimum Spanning Tree Problem Basic Concept of Minimum Spanning Tree Problem

Consider a connected graph G=(V, E), where

V={v1, v2, …, vn} or {1, 2, …, n} is a finite set of nodes E={(i,j) | i,j∈V} is a finite set of edges representing connections between these vertices.

Each edge has a positive real number denoted by W={w12, w13, …, wn-1,n} representing distance or cost.

1

1 1

1

1 1

1

1

min ( )

s. t . 1 : connectivity

| | 1, \{1},| | 2 : no loop

0 or 1, , 1,2, ...,

n n

ij iji j i

n n

iji j i

n n

iji S j S

j

ij

f w x

x n

x S S V S

x i j n

x 1 2

6

8

37 9

5 4

w23=4 ,x23=1

w12=2 ,x12=1

w18=3 ,x18=1

w67=5 ,x67=1

w49=8 ,x49=1

w79=4 ,x79=1

w78=1 ,x78=1

w56=6 ,x56=1

Fig. 7.2 Simple MST Model


7. Minimum Spanning Tree Problem In a real-life network optimization situation, the problem often requires s

atisfying additional constraints. Capacitated MST

Kershenbaum, A.: Computing capacitated minimal spanning trees efficiently, Networks, vol.4, pp.299–310, 1974.

Degree-constrained MST Narula, S. & C. Ho: Degree-constrained minimum spanning tree, Computers and Operations

Research, vol.7, pp.239-249, 1980. Stochastic MST

Ishii, H., H. Shiode, & T. Nishida: Stochastic spanning tree problem, Discrete Applied Mathematics, vol.3, pp.263-273,1981.

Quadratic MST Xu, W.: Quadratic minimum spanning tree problems and related topics, Ph. D. dissertation,

University of Maryland, 1984. Probabilistic MST

Bertismas, D.: The probabilistic minimum spanning tree problem, Networks vol.20, pp.245-275, 1990.

Multicriteria MST Zhou, G. & M. Gen: The genetic algorithms approach to the multicriteria minimum spanning

tree problem, in Kim, J. H., X. Yao, & T. Furuhashi Ed.: Proc. of the First Asia-Pacific Conference on Simulated Evolution and Learning, pp.387-394, Taejon, 1996.

Leaf-constrained MST Fernandes, L. M. & L. Gouveia: Minimal spanning trees with a constraint on the number of l

eaves, European J. of Operational Research, vol.104, pp.250-261, 1998.


7. Minimum Spanning Tree Problem Some researchers used genetic algorithms to deal with th

em. Probabilistic MST

Abuali, F., R. Wainwright, & D. Schoenefeld: A new encoding scheme for spanning trees applied to the probabilistic minimum spanning tree problem, in Proc. of the 5th Inter. Conf. on Genetic Algorithms, San Francisco, pp.470-475, 1995.

Palmer, C. C. & A. Kershenbaum: An approach to a problem in network design using genetic algorithm, Networks, vol. 26, pp.151-163, 1995.

Degree-constrained MST Zhou, G. & M. Gen: Approach to degree-constrained minimum spanning tree pro

blem using genetic algorithm, Engineering Design and Automation, vol.3, no,2, pp.157-165,1997.

Zhou, G. & M. Gen: A note on genetic algorithm approach to the degree-constrained spanning tree problems, Networks, vol.30, pp.105-109,1997.

Capacitated MST Chandy, K. M. & T. Lo: The capacitated minimum spanning tree, Networks, vol.

3, pp. 173-182, 1973. Zhou, G. & M. Gen: Approach to degree-constrained minimum spanning tree pro

blem using genetic algorithm, in Gen, M. & R. Chen, Genetic algorithms & Engineering Design, John Wiley, New York, 2000.


7. Minimum Spanning Tree Problem Quadratic MST

Zhou, G. & M. Gen: An effective genetic algorithm approach to the quadratic minimum spanning tree problem, Computers and Operations Research, vol.25, no.3, pp.229-247,1998.

Knowles, J. & D. Corne: A new evolutionary approach to the degree-constrained minimum spanning tree problem, IEEE Transactions on Evolutionary Computation, vol.4, no.2, pp.125-134, 2000.

Multicriteria MST Zhou, G. & M. Gen: Genetic Algorithm Approach on Multicriteria Minimum Span

ning Tree Problem, European J. of Operational Research, vol.114, pp.141-151, 1999.

Gen, M., G. Zhou & M. Takayama: Matrix-based Genetic Algorithm Approach on 　 Bicriteria Minimum Spanning Tree Problem with Interval Coefficients, J. of Japan Society for Fuzzy Theory and Systems, vol. 10, no.6, pp.643-656, 2000.

Zhou, G., H. Min & M. Gen: A genetic algorithm approach to the bi-criteria allocation of customers to warehouses, Int. J. of Production Economics, vol.86, pp.35-45, 2003.

Leaf-constrained MST Zhou, G. & M. Gen: Leaf-constrained spanning tree problem with genetic algorit

hms approach, Beijing Mathematics, vol.7, no.2, pp.50-62, 1998. Fernandes, L. M. & L.Gouveia: Minimal spanning trees with a constraint on the

number of leaves, European J. of Operational Research, vol.104, pp.250-261, 1998.



1. Multicriteria Minimum Spanning Tree (mc-MST)

1.1 Basic Concept of mc-MST 1.2 Genetic Algorithms Approach 1.2.1 Reviewing Encoding Methods

1.2.2 Genetic Representation 1.2.3 Genetic Operators

1.3 GA procedure for mc-MST1.4 Numerical Experiments

2. Degree-constrained Minimum Spanning Tree

3. Degree-based Permutation GA for dc-MST

4. Leaf-Constrained Minimum Spanning Tree


Multicriteria Minimum Spanning Tree (mc-MST) problem is of high

importance in practical network optimization.

A new approach to solve the mc-MST problem by using the Genetic Algorithms (GAs) is presented.

A skillful encoding for trees, denoted as Prüfer Number , is adopted for the GAs operation.

Combined with some MultiCriteria Decision Making (MCDM) techniques, the proposed GAs approach can :

(1) get the Pareto optimal solutions close to its ideal point as much as 　　 possible or;

(2) maintain all Pareto optimal solutions along the Pareto frontier.

1. Multicriteria Minimum Spanning Tree

Zhou, G. & M. Gen: Genetic algorithm approach on Multicriteria minimum spanning tree problem, European J. of Operation Research, vol. 114, pp. 141-152, 1999.

Chankong, V. & Y. Y. Haimes: Multiobjective Decision Making Theory and Methodology, North-Holland, New York, 1983.


The Multicriteria Minimum Spanning Tree (mc-MST) problem which has q objective and n nodes can be formulated as follows:

11

11 1

1

1 1

1

1 1

1

1

min ( )

min ( )

s. t . 1 : connectivity

| | 1, \{1},| | 2 : no loop

0 or 1, , 1,2, ...,

n n

ij iji j i

n nq

q ij iji j i

n n

iji j i

n n

iji S j S

j

ij

w xz

w xz

x n

x S S V S

x i j n

x

x

1.1 Basic Concept of mc-MST mc-MST Formulation


1.1 Basic Concept of mc-MST A basic understanding of Multiplecriteria Decision Making (MCDM) ca

n be illustrated by the following definition:

Definition 1: Given a set of feasible solution ,　　　　 solution is denoted as the Pareto optimal solution or

nondominated solution to the problem (1) if and only if there is no any other solution , satisfying the following conditions:

TX xx

Xx

'x

( ) ( ')l lz zx x

)'()( xx zz kk lk

},,2,1{ ql , for some

, for all

Definition 2: The point in objective space of the problem (1) is denoted as the ideal point, where

))(,),(),(()(21

xxxx zzzz p

)()( min xxx

zz kT

k

qk ,,2,1


Definition 3: In optimizing each objective of the problem (1),there are respectively q optimal solution and their corresponding objective values:

, 1, 2, ,j j qx

1,2, ,j q ( ), 1,2, , ,j jk k

k qz z x

This hyperplane is denoted as adaptive objective evaluation hyperplane to the problem (1)

Through these p points , there is a hyperplane which satisfies: where and are the solution of the following equations:

zkk

1

1

1

0, 1,2, ,

q

kk

qj

k kk

j qz

( , , )( , , , )1 2 1 2j j

k

j

z z z j p

),,2,1( pkk

1.1 Basic Concept of mc-MST


Operated on the coding of solution set of the problem to be solved at hand, instead of the solutions themselves.

Operated on a set of candidate solutions of the problem instead of a single one.

Operated not wholly randomly but including both the stochastic search (Crossover and Mutation) and the directed search (Evaluation and Selection) in Optimization.

Operated with a survival of the fittest mechanism

which can enforce the search got to the optimal solution.

1.2 Genetic Algorithms Approach

Zhou, G. & M. Gen: Genetic Algorithm Approach on Multicriteria Minimum Spanning Tree Problem, European J. of Operational Research, vol.114, pp.141-151, 1999.


For the MST problem, how to encode a tree is critical for GAs pproach.

Typical encodings for representing a tree can be classified as follows:

1.2.1 Reviewing Encoding Methods Tree Encodings

Edge Encoding: Piggott, P. & F. Suraweera: Encoding graphs for genetic algorithms: An investigation using the minimum spanning tree problem, in X. Yao, ed., Progress in Evolutionary Computation, Springer-Verlag, Berlin, pp.305-314, 1995.

Edge Encoding

Vertex Encoding Prüfer number Encoding:

Zhou, G. & M. Gen: Genetic algorithm approach on Multicriteria minimum spanning tree problem, European J. of Operational Research, vol.114, pp.141-152, 1999.

Edge and Vertex Encoding Link and Node biased Encoding:

Palmer, C. C. & A. Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, vol.26, pp. 151-163, 1995.

Degree-based Permutation Encoding: Zhou, G. & M. Gen: Approach to degree-constrained minimum spanning tree problem using genetic algorithm, in Gen, M. & R. Chen, Genetic algorithms & Engineering Design, John Wiley, New York, 2000.


1.2.1 Reviewing Encoding Methods Edge Encoding

The edge encoding is really an intuitive representation for a tree.Each element of the chromosome represents a possible edge in the graph so there are n(n-1)/2 edges. where n is the number of vertices. The value of each element represents whether the specific edge connects with the pair of nodes or not. A bit string can represent a candidate solution by indicating which edges are used 　　 in a spanning tree as illustrated.

x67x57x56x47x45x37x34x23 x27x17x16x12

x232 3

6

41 7

5x16

x67

x17

x56

x12

x45

x57

x47

x37x27 x34

0 0 1 1 1 0 0 1 0 1 1 0

Piggott, P. & F. Suraweera: Progress in Evolutionary Computation, Springer-Verlag ,1995Fig. 7.3 Illustration of Edge Encoding


cjk ’ = cjk + p1 bjk cmax + p2 (bj + bk) cmax

c11’ c12 ’ c1m ’

c21 ’ c22 ’ c2m ’

cm1 ’ cm2 ’ cmm ’

...

...

...

... ... ...

genetic representation

1.2.1 Reviewing Encoding Methods Link and Node biased Encoding

In this encoding, It holds a bias value for each node and each edge. Each node bias and each edge bias are an integer in the range from 0 to 255. The spanning tree corresponding to the encoding is found by running Prim’s MST algorithm on a modified cost matrix C’=[cjk’]

Palmer, C. & A. Kershenbaum: Networks, vol. 26, 1995

max

: Numner of nodes

: The minimum edge cost in the netowork

: The node bias associated with node

: The edge bias associated with edge

from node to node

, : Control paramers1 2

j

jk

m

c

b j

b

j k

p p


2 3

6

41 7

5143

138

12

26

89

823

172

6834 215

21

A chromosome represented byA chromosome represented byLink and Node Biased EncodingLink and Node Biased Encoding

by Prim’s algorithmby Prim’s algorithm

1.2.1 Reviewing Encoding Methods Link and Node biased Encoding

Fig. 7.4 Illustration of Link and Node Biased Encoding

--

8989

MM

MM

MM

143143

1212

8989

--

2121

MM

MM

MM

3434

MM

2121

--

215215

MM

MM

7676

MM

MM

215215

--

88

MM

154154

MM

MM

MM

88

--

2626

3434

143143

MM

MM

MM

2626

--

7676

1212

3434

6868

172172

2323

138138

--

764 5321

1

2

3

4

5

6

7


1.2.1 Reviewing Encoding Methods

Prüfer number 2 5 6 8 2 5

Prüfer number is adopted to represent all candidate solutions of the problem.

Any tree with n nodes, the encoding length is only n - 2

Prüfer number Encoding

Zhou,G. & M. Gen: Networks, vol. 30, 1997

Fig. 7.5 Illustration of Prüfer number Encoding

i j

1

7

2 5

3

8

6 4

w12=3

w27=5

w25=7 w58=2

w68=4

w46=6

w35=6(i, j) wij

(1, 2) 3

(2, 7) 5

(2, 5) 7

(3, 5) 6

(5, 8) 2

(6, 8) 4

(4, 6) 6

data set

wij

1 2 3 4 5 6



1

3 42

5 6 7 8

1 2 5 6 3 4 7 8

3 3 1 1 1 3 1 1

Tree

node ID is node number based on the depth first search (DFS),degree at node ID is the number of connecting nodes.

node ID j

degree yj at node ID j

w12=5w13=3 w14=6

w25=4 w26=3 w47=3 w48=5

(i, j) wij

(1, 2) 5

(1, 3) 3

(1, 4) 6

(2, 5) 4

(2, 6) 3

(4, 7) 3

(4, 8) 5

data set

Degree-based Permutation Encoding

Zhou, G. & M. Gen: Genetic algorithms & Engineering Design, John Wiley, 2000

Fig. 7.6 Illustration of Degree-based Permutation Encoding

1 2 3 4 5 6 7 8Degree-based permutation


1.2.2 Genetic Representation Procedure of Encoding for Prüfer number (Zhou & Gen, 1997)

input: spanning tree data set T

output: Prüfer number P

step 1: Let node i be the smallest labeled leaf node in a labele

d tree T.

step 2: Let j be the first digit in the encoding, as the code j

incident to i is uniquely determined. The encoding is

built by appending digits from left to right.

step 3: Remove node i and the link from i to j, thus there is a

tree with k-1 nodes.

step 4: Repeat the steps above until one link is left. P is obtain

ed.

input: spanning tree data set T

output: Prüfer number P

step 1: Let node i be the smallest labeled leaf node in a labele

d tree T.

step 2: Let j be the first digit in the encoding, as the code j

incident to i is uniquely determined. The encoding is

built by appending digits from left to right.

step 3: Remove node i and the link from i to j, thus there is a

tree with k-1 nodes.

step 4: Repeat the steps above until one link is left. P is obtain

ed.


1.2.2 Genetic Representation Procedure of Decoding for Prüfer number (Zhou & Gen, 1997)

input: Prüfer number P

output: spanning tree data set T

step 1: Let P be the original Prüfer number and let P' be the set of all

nodes not included in P, which are designated as eligible

nodes for consideration in building a tree T.

step 2: Let i be the eligible node with the smallest label. Let j be the

left most digit of P. Add the edge from i to j into the tree T.

Remove i from P' and j from P. If j does not occur anywhere

in P, put it into P'. Repeat the process until no digits are left in

P.

step 3: If no digits remain in P, there are exactly two nodes, r and s,

still eligible for consideration. Add a link from r to s into tree

and form a tree with k-1 links.

input: Prüfer number P

output: spanning tree data set T

step 1: Let P be the original Prüfer number and let P' be the set of all

nodes not included in P, which are designated as eligible

nodes for consideration in building a tree T.

step 2: Let i be the eligible node with the smallest label. Let j be the

left most digit of P. Add the edge from i to j into the tree T.

Remove i from P' and j from P. If j does not occur anywhere

in P, put it into P'. Repeat the process until no digits are left in

P.

step 3: If no digits remain in P, there are exactly two nodes, r and s,

still eligible for consideration. Add a link from r to s into tree

and form a tree with k-1 links.


Fig. 7.7 Illustration of crossover operation (One-cut Point Crossover)

1.2.3 Genetic Operators Crossover and Mutation

Prüfer number encoding can always represent a tree after any crossover or mutation operations, simply the uniform crossover operator is adopted. Mutation is performed as random perturbation within the range from 1 to n.

Fig. 7.8 Illustration of mutation operation (Altering Mutation)


As to selection procedure, the -selection is adopted.( )

1.2.3 Genetic Operators Evaluation and Selection (Strategy I)

input: chromosomes

output: fitness values of each chromosome

step 1: Decode all chromosomes and calculate their objective

values in each objective function.

step 2: Determine the fitness value eval(T) of all chromosomes

according to the following formula:

2

1

λ)(k

kk fTeval

ts.coefficienweightingthearewhere )2,1(λ kk


input: chromosomesoutput: fitness values of each chromosomestep 1: Determine all nondominated chromosomes and assign a large dummy fitness value to them.step 2: Calculate each chromosome’s niche count ;

, where

step 3: Calculate the shared fitness value of each chromosome by dividing its dummy fitness by its niche count.

step 4: Ignore all sorted nondominated chromosomes, go to step 1 and continue the process.

cP

jm

j jkk

m dshcP

( )

sh jkdd

d

jk

share

( ) ( ) 12

jk shared d sh jkd( ) 0, if ; otherwise

For selection operation roulette wheel is adopted.

1.2.3 Genetic Operators Evaluation and Selection (Strategy II)


procedure: GA for mc-MSTinput: network data (V, A, W), associated number set in each edge, GA parametersoutput: Pareto optimal solutions E(P,C)begin

t ←0;initialize P(t) by Prüfer number encoding;objectives z1(P), z2(P),…, zq(P);

create Pareto E(P)fitness eval(P) by Prüfer number decoding;while (not termination condition) do crossover P(t) to yield C(t) by uniform crossover operator; mutation P(t) to yield C(t) by random perturbation; objectives z1(C), z2(C) ,…, zq(C);

update Pareto E(P,C) if (not preference condition) fitness eval(P,C) by strategy I; else fitness eval(P,C) by strategy II; select P(t+1) from P(t) and C(t) by roulette wheel selection; t ← t+1; endoutput Pareto optimal solutions E(P,C);

end

1.3 GA procedure for mc-MST GA procedure for Multicriteria Minimum Spanning Tree (mc-MST)


Two networks with 10-vertex and 50-vertex are tested by the proposed GAs approach. The weights defined on two objectives are randomly generated and respectively distributed over [0, 100] and [0, 50].

The parameters for the GAs operation are set as follows:

population size popSize = 200 crossover probability pC = 0.2 mutation probability pM = 0.05 maximum generation maxGen = 500

1.4 Numerical Experiments

Extreme Pareto optima Problem Size (Number of nodes)

Ideal point (z1

o, z2o) (z1

o, z2) (z1, z2o)

10 50

(156, 112) (583, 513)

(156, 195) (583, 1433)

(387, 112) (2366, 513)

Table 7.1 Ideal points and extreme Pareto optimal solutions


Fig. 7.9 Illustration of GAs approach on the mc-MST by Strategy I


Numerical Experiments - Result


Fig. 7.10 Illustration of GAs approach on the mc-MST by Strategy II


(a) 10-vertex network (b) 50-vertex network

Numerical Experiments - Result



1. Multicriteria Minimum Spanning Tree (mc-MST)


2.1 Basic Concept of dc-MST2.2 Genetic Algorithms Approach2.3 GA procedure for dc-MST



4. Leaf-constrained Minimum Spanning Tree


Degree-constrained Minimum Spanning Tree (dc-MST) problem is of high importance in practical network design.

The dc-MST problem is NP-hard, no effective algorithms exitto solve this problem.

A new approach to solve the dc-MST problem by using the Genetic Algorithms (GAs) is presented.

A skillful encoding for trees, denoted as Prüfer Number, is adopted for the GAs operation.

Numerical examples are suggested to test the effective-ness of the proposed GAs approach compared with the results by existing heuristics and their Lower Bound.


Description of Degree-constrained Minimum Spanning Tree


2.1 Basic Concept of dc-MST

3

4 56

8 9

1

A spanning tree with degree dj = 3

Number of degrees:y4= 4 case

46

8 9

1

5

2

3

2

Fig. 7.11 A spanning tree with degree

y8=1≤ d8 y9=1≤ d9

y7=3≤ d7

y6=1≤ d6 y4=4>d4y5=1≤ d5

y7=3≤ d7

y3=1≤ d3

y1=1≤ d1

y8=1≤ d8 y9=1≤ d9

y7=3≤ d7

y6=1≤ d6 y5=1≤ d5

y7=3≤ d7

y3=1≤ d3

y1=1≤ d1

7 7

y4=3≤d4

yj ： Degree value on vertex j

Number of degrees:y4= 3 case



Notations Indices

i, j : the index of node, i, j =1, 2, …, n

Parametersn: the number of nodes in the network


d ｊ : the constrained degree value on vertex ｊ


cij : the cost of connecting node i to node j, i.e., the cost of

link (i,j); the cost matrix (cij) is symmetric.

yj ： degree value on vertex

Decision Variable



If we assume that there is a degree constraint on each vertex such that, at each vertex , the degree value yj is at most a given value , then the problem can be formulated as follows:

jvjd


n

…

3

j

1 2

wjn ,xjn=1

…

…

w3n ,x3n=0

w12 ,x12=1

…

…

…

…

y1=1≤ d1 y2=2 ≤ d2

y3=1 ≤ d3

yj=3 ≤ dj…

yn=1 ≤ dn

Fig. 7.12 Simple dc-MST Model

1

1 1

1

1 1

1

1

min

s. t . 1 :

| | 1, \{1},| | 2 :

, 1, , :

0 or 1, , 1, 2, ...,

, 1, 2,...,

n n

ij iji j i

n n

iji j i

n n

iji S j S

j

j j

ij

j

z c x

x n

x S S V S

y d j n

x i j n

y I j n

connectivity

no loop

degree constriant

Mathematical Model of dc-MST Problem



Fig. 7.13 A spanning tree with degree

3

6

5

48

9

1

Number of degrees:y6= 4

2

y8=2≤ d8

y9=1≤ d9 y7=1≤ d7

y4=1≤ d4y6=4>d6

y5=1≤ d5

y3=1≤ d3y1=2≤ d1

7

y2=3≤ d2

3

6

7

48

9

1 2

y8=2≤ d8

y9=1≤ d9

y7=1≤ d7

y4=1≤ d4y6=3>d6

y5=2≤ d5

y3=1≤ d3y1=2≤ d1

5

y2=3≤ d2

Repairing illegal Chromosome

Number of degrees:y6= 3

Sample : A spanning tree with degree di = 3

repair

Case: Case:


Let be the set of vertices whose degree has not been checked and repaired in a chromosome. If a vertex violates the degree constraints (degree value d ), this means that the number of this vertex in the chromosome is more than d - 1, then decrease the number of the vertex as illustrated as follows:

Fig. 7.14 Illustration of degree modification on a chromosome

dcV


Repairing illegal Chromosome

2 6 6 2 8 1offspring 3

2 2 8 1parent 6 6 6

check the extra vertex

replace with a digit from dcV


One-cut point crossover and altering mutation are used The selection operation consists of both - selection and roulette wheel selection.

procedure: Selectioninput: P(t), C(t), ,output: P(t+1)begin select best different chromosomes; if then select chromosomes by roulette wheel selection; output P(t+1);end

' '

'

( )


Genetic Operations

'


Convert each chromosome into a tree in the form of edge set according to the Prüfer number decoding procedure.

Calculate the fitness value for each chromosome in the form of total cost of the tree according to the objective in the dc-MST Problem.


Evaluation


procedure: GA for dc-MST

input: network data (V, A, W), associated number set in each edge, GA parameters

output: best MST solution

begin

t ←0;

initialize P(t) by Prüfer number encoding;

check P(t) degree constraint and repairing P(t);

fitness eval(P) by Prüfer number decoding;

while (not termination condition) do

crossover P(t) to yield C(t) by one-cut point crossover;

mutation P(t) to yield C(t) by altering mutation;

check C(t) degree constraint and repairing C(t);

fitness eval(C) by Prüfer number decoding;

select P(t+1) from P(t) and C(t) by roulette wheel and ( +)-selection;

t ← t+1;

end

output best MST solution;

end

2.3 GA procedure for dc-MST


The numerical example was given by Savelsbergh and Volgenant who solved it by using heuristic algorithm denoted as edge exchange. It is a 9-vertex completer undirected network and optimal solution is 2256.


Savelsbergh, M. & T. Volgenant: Edge exchanges in the degree- constrained spanning tree problem, Computers & Operations Research, vol. 12, no. 4, pp. 341-348, 1985.

1 - 224 224 361 671 300 539 800 943

2 - 200 200 447 283 400 728 762

3 - 400 566 447 600 922 949

4 - 400 200 200 539 583

5 - 600 447 781 510

6 - 283 500 707

7 - 361 424

8 - 500

9 -

node i 1 2 3 4 5 6 7 8 9

Table 7.2 The numerical example by Savelsbergh and Volgenant


The parameters for the GAs operation are set as follows: popSize = 300; crossover probability pC = 0.2; maxGen = 500; mutation probability pM = 0.2;

Fig. 7.15 Solution distribution for the dc-MST using GAs

2.4 Numerical Experiments Numerical Experiments – Results Analysis


Fig. 7.16 Solution comparison between two selection strategies




Problems Size

(Number of nodes) LB(without) GA(with dc)

Degree minVal % minVal

10

20

30

40

50

5(2)

4(2)

4(4)

7(3)

6(4)

117

233

316

419

513

5.13

1.72

4.48

7.16

7.99

123

237

327

449

554

LB: lower bound without degree constraint;GA: GA for dc-MST ; minVal: minimal value; percentage of results by GA to LB;

Numerical Experiments – Results

Table 7.3 Comparison between results by the LB and GA






3.1 Concept on Degree-based Permutation GA 3.2 Genetic Algorithms Approach 3.2.1 Genetic Representation 3.2.2 Genetic Operators

3.3 Degree-based Permutation GA for dc-MST3.4 Numerical Experiments



Spanning tree is a basic topology structure in network

design problems like transportation, telecommunication

and distribution system.

Degree-constrained minimum spanning tree (dc-MST)

is a more realistic representation in practice but it is a

NP-hard problem.

A new encoding is developed to deal with this problem

by using Genetic Algorithms (GAs).



Simple Heuristic Approach: Narula and Ho (Computer & Operations Research, vol. 7, 1980)

Edge Exchange Heuristic Approach: Savelsbergh and Volgenant (Computers & Operations Research, vol. 12, no. 4, 1985)

Langrangean Multipliers Approach: Volgenant (European J. of Operational Research, vol. 39, 1989)

Genetic Algorithm Approach using Prüfer Number: Zhou and Gen (Proceedings of 1996 IEEE Internat. Conference on System, Man and Cybernetics, 1996, and Networks, vol. 30, 1997)

The dc-MST problem was first proposed by Narula and Ho, regarded

as NP-hard problem (1980).

3.1 Concept on Degree-based Permutation GA Literature Review


Any Genetic Algorithms approach designed for a

particular problem should address two main factors:

(1) genetic representation of solutions to the problem, usually termed as chromosome or individual.

(2) genetic operators that would alter the genetic composition of chromosomes during the evolutionary process.

3.1 Concept on Degree-based Permutation GA



One of the classic theorems in graphical enumeration is Cayley's theorem that there are n(n-2) distinct labeled trees on a complete network with n vertices.

Prüfer provided a constructive proof of Cayley's theorem by establishing a one-to-one mapping between such trees and the set of all string of n-2 digits.

This means that it is possible to use only n-2 digits permutation to uniquely represent a tree where each digit is an integer between 1 and n inclusive.

This permutation is usually known as Prüfer number. Prüfer, H.: Neuer Beweis eines Satzes über Permutationen, Archiv

fuer Mathemtische und Physik, vol. 27, pp. 742-744, 1918. In this study, Prüfer number is used to represent a candidate tree an

d initial population is generated using complete random method.


3.2 Genetic Algorithms Approach Shortcoming of the Prüfer number

The Prüfer number has the great possibility to lose fit structure evolved in

the evolutionary process. Above figure shows that changing even one digit

of a Prüfer number can change the tree dramatically.

Fig. 7.17 Illustration of Prüfer number

3 4

5

612

Prüfer number 3 3 4 1

3 4

5

621


Lo, C. C. & W. H. Chang: ”A Multiobjective Hybrid Genetic Algorithm for the Capacitated Multipoint Network Design Problem”, IEEE Transactions on Systems, Man, and Cybernetics vol.30, no.3, pp.461-470, 2000.


3.2 Genetic Algorithms Approach Shifting Prüfer Vector

Shifting Prüfer vector maintains maximum locality; i.e., it keeps the similarity between chromosomes. In most cases, the difference is only one edge by this operator.

Fig. 7.18 Illustration of Prüfer number

3 4

5

612


4 3

5

612


input: parentoutput: offspring step 1: Randomly select the nonleftmost element of a Prüfer vector.step 2: Replace the leftmost element of the same vector by nonleftmost element.

select randomly


3.2.1 Genetic Representation

An intuitive idea is to use a two-dimension structure to encode a spanning

tree with degree constraint, denoted as tree-based permutation:

one dimension for the connection among vertices

the other dimension for the degree value of each vertex

It needs a matrix to represent a chromosome for a n vertex tree. The

genes in the degree dimension take inclusively the integers from 1 to b the

constrained degree value of all vertices. This representation was denoted as

degree-based permutation by Zhou and Gen.

Zhou, G. & M. Gen: A new tree encoding for the degree-constrained

spanning tree problem, in Gen, M. & R. Chen, Genetic algorithms &

Engineering Design, John Wiley, New York, 2000.

For an undirected tree, we can take any vertex as the root vertex. All other

vertices are then regarded as being connected to it hierarchically.

n2



input: tree data set

output: chromosome

step 1: Select any vertex (root vertex) in a labeled tree T, put it as the first

digit in the vertex dimension of the permutation and its degree

value as the first digit in the degree dimension, and let that vertex be

the current vertex.

step 2: Check the successor vertex of the current vertex from left branch

to right branch. If there is a successor vertex, put it into the vertex

dimension and its degree value into the degree dimension, and then

go to step 3. If there is no such vertex, let the predecessor vertex be

the current vertex, and return to step 2.

step 3: If the successor vertex is not a leaf vertex, let the successor vertex

be the current vertex, then go to step 2. If the successor vertex turns

to be a leaf vertex, delete it and go to step 4.

step 4: If all vertices have been checked, stop; otherwise, go to step 2.

Encoding Procedure for Degree-based Permutation (Zhou & Gen, 1997)



input: chromosome

output: tree data set

step 1: Set k ← 1 and j ← 2.

step 2: Select the vertex in P1(k), say vr where r= P1(k), select the vertex in P1 (j), say vs where s= P1 (j), add the first edge from r to s into a tree.

step 3: Let P2 (j) ← P2(j) - 1.

step 4: If P2(j) ≥ 1, let k ← j and j ← j+1.

step 5: If k < 1, stop.

step 6: If P2(k) ≥ 1, select the vertex in P1(k), say vr where r = P1(k), and let P2(k) ← P2(k) - 1; otherwise, let k ← k-1 and go to step 5.

step 7: If P2(j) ≥ 1, select the vertex in P1(j), say vs where s = P1(j), and let P2(j) ← P2(j) - 1; otherwise, let j ← j+1 and go to step 6.

step 8: Add the edge from r to s into the tree and go to step 4.

Decoding Procedure for Degree-based Permutation (Zhou & Gen, 1997)



1

3 42

5 6 7 8

1 2 5 6 3 4 7 8

3 3 1 1 1 3 1 1

Tree:

Degree-based permutation:

node ID is node number based on the depth first search (DFS),degree at node ID is the number of connecting nodes.

vt1 :node ID j

vt2 :degree yj at node ID j

w12=5w13=3 w14=6

w25=4 w26=3 w47=3 w48=5

(i, j) wk

(1, 2) 5

(1, 3) 3

(1, 4) 6

(2, 5) 4

(2, 6) 3

(4, 7) 3

(4, 8) 5

data set

1 2 3 4 5 6 7 8

Example of Degree-based Permutation Encoding


For the initial population, each chromosome can be generated randomly, but in order to keep the connectivity between nodes, the genes in the degree dimension need to satisfy the following conditions:

(1) the degree value for the current node in degree dimension should hold: no less that 1 and no greater than d ( = 3).

(2) the degree value for the current node together with the number of the rest nodes should hold: no less than drest and no greater

than 2(n - 1) - dused

where drest is the total lower bound of the degree values for all those nodes whose degree value in P2 has not been assigned and dused

is the total degree value of the nodes whose degree value in P2 has been assigned.

3.2.1 Genetic Representation Initial Population


Three mutation operators are adopted to improve the chromosomes in the evolutionary process:

(1) Exchange Mutation

(2) Inversion Mutation

(3) Insertion Mutation

3.2.2 Genetic Operators


3.2.2 Genetic Operators Exchange Mutation on Vertices

input: parent output: offspringstep 1: Selects two positions at random

step 2: Exchanges the values on these positions

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

3 3 1 1 1 3 1 1 3 3 1 1 1 3 1 1

1

3

5 6 7 8

1

3

5 6 7 8

Exchange Mutation

42 24

Fig. 7.18 Illustration of Exchange mutation

node ID j



3.2.2 Genetic Operators Insertion Mutation

input: parentoutput: offspringstep 1: Selects a string of genes (branch) at random. step 2: Inserts it in a random gene (vertex).

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

3 3 1 1 1 3 1 1 3 3 1 2 1 1 2 1

6

insertion point

1

43

5 6 7 8

2

1

4

7

5 8

2 3

Fig. 7.19 Illustration of Insertion mutation

node ID j



3.2.2 Genetic Operators Inversion Mutation

input: parentoutput: offspringstep 1: Selects a substring of genes (branch) at random.

step 2: Inverts the substring between these two gene (vertex).

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

3 3 1 1 1 3 1 1 3 3 1 1 1 3 1 1

6

select substring

1

43

5 6 7 8

2

1

4 25

7 83

Inverse Mutation

Fig. 7.20 Illustration of Exchange mutation

node ID j



The evaluation procedure can be described as follows:

(1) Convert a chromosome into a tree according to the

decoding procedure;

(2) Calculate the total cost of a tree according to the

objective function and take the reciprocal value of it as

the fitness value of that chromosome.

(3) Repeat the procedure on all chromosomes.


Evaluation


As to selection, we adopt the ( +)-selection strategy.

But in order to avoid the premature convergence of the

evolutionary process, our selection strategy only selects

different best chromosomes from parents and offspring.

If there are no different chromosomes available, the vacant

pool of population is filled with renewal chromosomes generated

in the same way as the initial population does.


Selection


3.3 Degree-based Permutation GA for dc-MST

procedure: Degree-based Permutation GA for dc-MST

input: network data (V, A, W), degree set, GA parameters

output: best MST solution

begin

t ←0;

initialize P(t) by degree-based permutation encoding;

fitness eval(P) by degree-based permutation decoding;

while (not termination condition) do

mutation P(t) to yield C(t) by exchange, inversion and insertion;

fitness eval(C) by degree-based permutation decoding;

select P(t+1) from P(t) and C(t) by ( +)-selection;

t ← t+1;

end

output best MST solution;

end


The numerical example on the dc-MST problem is a nine-node

complete network shown in Table 7.4. It was given by Savelsbergh and Volgenant who solved it using heuristic algorithm denoted as edge exchanges and the optimal solution is 2256.

Table 7.4 Edge weights of the nine-node dc-MST problem


1 - 224 224 361 671 300 539 800 943

2 - 200 200 447 283 400 728 762

3 - 400 566 447 600 922 949

4 - 400 200 200 539 583

5 - 600 447 781 510

6 - 283 500 707

7 - 361 424

8 - 500

9 -

node i 1 2 3 4 5 6 7 8 9


The parameters for the proposed GA approach with degree- based permutation (dpGA) are set as:

population size popSize = 100; mutation probability for exchange pM = 0.2; mutation probability for inversion pM = 0.2; mutation probability for insertion pM = 0.6; maximum generation maxGen = 500;

The parameters for the GA approach with Prüfer number (pnGA) are set as:

population size popSize = 100; crossover probability pC = 0.5;

mutation probability pM = 0.01; maximum generation maxGen = 500;

The constrained degree value for all nodes d = 3, and run by 20 times.

3.4 Numerical Experiments Parameter Setting


Fig. 7.21 Evolutionary process by two encodings on the dc-MST


Numerical Experiments – Results Analysis


Fig. 7.22 Illustration of the sensitivity on mutation operations



mutation probability

(crossover probability = 0.0)


Fig. 7.23 Illustration of the sensitivity on crossover operation


insertion probability= 0.6

exchange and inversion probability= 0.2

crossover probability


LB: lower bound without degree constraint; pnGA: GA with Prüfer number encoding; dpGA: GA with degree-based permutation; minVal: minimal value;%: percentage of results by GA to LB; CPU: CPU runtimes on average;

Table 7.5 Comparison between results by the LB and GAs

Problem LB pnGA dpGA

(Number of nodes) minVal degree minVal % CPU (s) minVal % CPU (s)

10 117 5(2) 123 5.13 12.2 123 5.13 8.2 20 233 4(2) 237 1.72 86.3 237 1.72 51.5 30 316 4(4) 327 3.48 250.6 321 1.58 134.6 40 419 7(3) 449 7.16 560.7 428 2.15 304.2 50 513 6(4) 554 7.99 1120.2 531 3.51 574.8



7. Minimum Spanning Tree Problems





4.1 Basic Concept of lc-MST


4.3 GA procedure for lc-MST



Spanning tree is a basic topology structure in network design problems like transportation, telecommunication and distribution systems.

Leaf-constrained minimum spanning tree (lc-MST) is a more realistic representation in practice but it is a NP-hard problem.

A new encoding is developed to deal with this problem by using Genetic Algorithms (GAs).


Description of Leaf-constrained minimum spanning tree

Zhou, G. & M. Gen: “A genetic algorithm approach on tree-like telecommunication network design problem”, J. of Operational Research Society, vol.54, pp.248-254, 2003.


4.1 Basic Concept of lc-MST Description of Computer Communication Network

Illustration of a network with spanning tree structure

Conc.

Conc.

Mux

.Mux.

Mux.

Conc.

Terminals

Conc.: concentrator

Mux.: multiplexer

Host Computer

Host Computer

Host Computer

. . .

.

.

.

Fig. 7.24 Illustration of a network with spanning tree structure



3

4

2

56

7

8 9

1

4

2

7

9

13

56

8

A spanning tree with leaves

Number of leaves Number of leaves

u8=1 u9=1

u5=1u6=1

u1=1

u7=0

u4=0

u2=0

u6=1

u9=1

u1=1u3=0

u7=0

u4=0

u2=0

u5=0

u8=0

u3=1

ui : the 0,1 decision variable; 1, if node i has degree 1 in the

solution, and 0, otherwise.

9

1

3ii

R u

9

1

6ii

R u

9

1

3ii

R u

Case:Case:


4.1 Basic Concept of lc-MST Notations

Indicesi, j : the index of node, i, j =1, 2, …, n

Parametersn: the number of nodes in the network


R : the number of terminals


cij : the cost of connecting node i to node j, i.e., the cost of link (i,j); the cost matrix

　　 (cij) is symmetric.

Decision Variables


uj : the 0,1 decision variable; 1, if node j has degree 1 in the solution,

　　 and 0, otherwise.



Mathematical Model of lc-MST

1

1 1

minn n

ij iji j i

z c x

vn

j

1 2

…

c1j ,x1j=0

c12 ,x12=1

…

…

…

u1=1

i

cij ,xij=1…

u2=0

c2i ,x2i=0

…

…

…

ui=1

uj=0

un=1

Fig. 7.25 Simple lc-MST Model

1

1 1

1

1

1

s. t . 1 :

| | 1, \{1},| | 2 :

1 1 , 1, 2, , :

0 or 1, , 1, 2, ...,

0 or 1 1,2,...,

n n

iji j i

n n

iji S j S

j

n

ii ijj

ij

j

x n

x S S V S

u x R i nu

x i j n

, j nu

connectivity

no loop

degree constriant


In applying the Prim’s algorithm to obtain a MST solution, if the number of all leaves are equal to the required leaf- constrained value k, this solution is definitely the optimal

solution to its lc-MST problem.

If we do not consider the leaf-constraints on it, this MST solution can be simply regarded as the lower bound of its corresponding lc- MST problem.


Estimation of Lower Bound


Firstly, obtaining the MST solution by using the Prim’s algorithm (usually the leaves in a MST solution is more that the required leaf-constrained value).

Secondly repairing the MST solution by decreasing

its number of leaves.

In modification, we only select an one-leaf subtree and append it on another leaf at the expense of increasing the total weights as less as possible.


Estimation of Upper Bound


One crossover and two mutation operators are adopted to improve the chromosomes in the evolutionary process:

Order Crossover : Only operated on node dimension, not together with the degree dimension.

(1)

Exchange Mutation : Selects two genes (nodes) at random and then swaps the gene.

(2)

Insertion Mutation : Selects a string of genes (nodes) at random and inserts it in a random gene.

(3)



Order Crossover

3 4 76 8521

1 3 11 1133

3 4 76 2518

1 3 11 1133

1 7 34 2856

1 3 11 1133

parent 1

offspring

parent 2

selected substringinput: parent chromosomeoutput: offspring chromosomestep 1: Select a substring from one parent at random.

step 2: Produce a proto-child by copying the sub-string into the corresponding positions of it.

step 3: Delete the cities which are already in the sub-string from the second parent. The result sequence of cities contains the cities that the proto-child needs.

step 4: Place the cities into the unfixed positions of the proto-child from left to right according to the order of the sequence to produce an offspring.

Fig. 7.26 Illustration of Order crossover



It is possible that there are such chromosomes whose leaves number is not equal to the leaf-constrained value.

Fig. 7.27 Illustration of leaves modification

4.2 Genetic Algorithms Approach Repairing on Leaves

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

3 3 1 1 1 3 1 1 3 3 1 1 2 1 2 1

1

3

5 6 7 8

1

3

5 6 7 8

42 42

leaf one-leaf subtree

input: parent chromosomeoutput: offspring chromosomestep 1: Selects one-leaf subtree at random

step 2: Insert it on another randomly selected one-leaf subtree.

node ID j



The evaluation procedure can be described as follows:

(1) Convert a chromosome into a tree according to the decoding procedure;

(2) Calculate the total cost of a tree according to the objective function and take the reciprocal value of it as the fitness value of that chromosome.

(3) Repeat the procedure on all chromosomes.


Evaluation


The ( +) - selection strategy is adopted in selection.

If there are no different chromosomes available, the vacant pool of population is filled with renewal chromosomes generated in the

same way as the initial population does.

In order to avoid the premature convergence of the evolutionary process, our selection strategy only selects different best chromosomes from parents and offspring.


Selection


procedure: GA for lc-MSTinput: network data (V, A, W), degree set, GA parametersoutput: best MST solutionbegin　　 t ← 0;　　 initialize P(t) by degree-based permutation encoding; check P(t) leaf constraint and repairing P(t); 　　 fitness eval(P) by degree-based permutation decoding;　　 while ( not termination condition ) do　　　　 crossover P(t) to yield C(t) by order crossover mutation P(t) to yield C(t) by exchange and insertion check C(t) leaf constraint and repairing C(t);　　　　 fitness eval(C) by degree-based permutation decoding;　　　　 select P(t+1) from P(t) and C(t) by ( +)-selection;　　　　 t ← t +1;　　 end output best MST solution;end

4.3 GA procedure for lc-MST GA procedure for Leaf-constrained minimum spanning tree (lc-MST)


The numerical examples on the lc-MST problem are

generated randomly. The cost matrix Cij is taken as the

integer uniformly and randomly distributed in the range of

[10,80].

In order to testify the effectiveness of the

proposed GA approach on the lc-MST problems,

all the GA’s results are compared with their lower

and upper bounds.



The analysis of the parameters setting for the proposed GA approach was tested on a ten-node lc-MST problem and the population size was fixed with 200, and found that:

Insertion mutation is more important than exchange mutation and order crossover.

4.4 Numerical Experiments Parameters Setting

Fig. 7.28 Illustration of the sensitivity on mutation and crossover

mutation probability mutation probability crossover probability


Exchange mutation probability: 0.2

Fig. 7.29 Evolutionary process by two encodings on the lc-MST

Insertion mutation probability: 0.8

Order crossover probability: 0.2

4.4 Numerical Experiments Better Parameters Setting

crossover probability

exchange mutation probability=0.2

insertion mutation probability=0.8


dj Lower Bound Upper Bound GAs Result

10 3 174(5) 291(3) 186

20 3 266(8) 283(3) 277

20 5 266(8) 277(5) 272

30 5 367(12) 421(5) 402

30 8 367(12) 414(8) 389



Table 7.6 Experimental ResultsProblem Scale

(Number of nodes)


dj (GA-LB)/LB (UB-GA)/UB CPU(sec.)

10 3 6.90% 2.62% 11.15

20 3 4.14% 2.12% 81.23

20 5 2.26% 1.18% 67.52

30 5 9.54% 2.14% 346.22

30 8 5.60% 6.04% 324.53

LB:lower bound;

UB:upper bound;



Table 7.7 Experimental ResultsProblem Scale

(Number of nodes)


Conclusion We have applied successfully the genetic algorithms

to solve kind of typical constrained spanning tree problems.

Those constrained spanning tree problems are all NP-hard combinatorial optimization problems which have some disadvantages. They are only effective on smaller scale problem They need some special mathematical skills They can not find the optimal or near-optimal solutions in

most cases To overcome those disadvantage

We developed a new tree encoding denoted as tree-based permutation for the GA approach on the mc-MST, dc-MST, and lc-MST problem.

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