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Chapter 32. Computational Performance of Three Subtour Elimination Algorithms for Solving Asymmetric Traveling Salesman Problems

Chapter 32. Computational Performance of Three Subtour Elimination Algorithms for Solving Asymmetric Traveling Salesman Problems

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496



T.H.C. Smith, V. Srinivasan, G.L. Thompson



traveling salesman problem these algorithms as well as another interesting algorithm of Bellmore and Malone [1] based on the 2-matching relaxation of the

symmetric traveling salesman problem are completely dominated in efficiency by

the branch-and-bound algorithm of Held and Karp [lo] (further improved in [S])

based on a 1-tree relaxation of the traveling salesman problem. In [13] an implicit

enumeration algorithm using a LIFO (Last I n First O u t ) depth first branching

strategy based on Held and Karp’s 1-tree relaxation was introduced and extensive

computational experience indicates that algorithm to be even more efficient than

the previous Held-Karp algorithms.

In [I71 Srinivasan and Thomspon showed how weak lower bounds can be

computed for the subproblems formed in the Eastman-Shapiro branch-and-bound

algorithm [5, 111. The weak lower bounds are determined by the use of cell cost

operators [14, 151 which evaluate the effects on the optimal value of the objective

function of parametrically increasing the cost associated with a cell of the

assignment problem tableau. Since these bounds are easily computable, it was

suggested in [I71 that the use of these bounds instead of the bounds obtained by

resolving or post-optimizing the assignment problem for each subproblem, would

speed up the Eastman-Shapiro algorithm considerably. In this paper we propose

and implement a straightforward LIFO implicit enumeration version of the

Eastman-Shapiro algorithm as well as two improved LIFO implicit enumeration

algorithms for the asymmetric traveling salesman problem. In all three of these

algorithms the weak lower bounds of [I71 are used to guide the tree search. The use

of weak lower bounds in the branch-and-bound subtour elimination approach is

explained with an example in [17].

We present computational experience with the new algorithms on problems of up

to 200 nodes. The computational results indicate that the proposed algorithms are

more efficient than (i) the previous subtour elimination branch-and-bound algorithms and (ii) a LIFO implicit enumeration algorithm based on the 1arborescence relaxation of the asymmetric traveling salesman problem suggested

by Held and Karp in [9], recently proposed and tested computationally in [12].



2. Subtour elimination using cost operators



Subtour elimination schemes have been proposed by Dantzig, et al. [3, 41,

Eastman [5], Shapiro [ I l l , and Bellmore and Malone [l]. The latter four authors

use, as we do, the Assignment Problem (AP) relaxation of the traveling salesman

problem (TSP) and then eliminate subtours of the resulting A P by driving the costs

of the cells in the assignment problem away from their true costs to very large

positive or very large negative numbers.

The way we change the costs of the assignment problem is (following [17]) to use

the operator theory of parametric programming of Srinivasan and Thompson [14,

151. To describe these let 6 be a nonnegative number and ( p , q ) a given cell in the



Computational performance of subtour elimination algorithms



497



assignment cost matrix C = {cij}. A positive (negative) cell cost operator SC&(SC,)

transforms the optimum solution of the original A P into an optimum solution of the

problem AP+(AP-) with all data the same, except

c ; = c,



+ 6 ; (c,=



c, - 6).



The details of how to apply these operators are given in [14, 151 for the general case

of capacitated transportation problems and in [17] for the special case of assignment problems. Specifically we note that p + ( p - ) denotes the maximum extent to

which the operator SCL(SC,) can be applied without needing a primal basis

change.

Denoting by Z the optimum objective function value for the AP, the quantity

( Z + p + ) is a lower bound (called a weak lower bound in [17]) on the objective

function value of the optimal AP-solution for the subproblem formed by fixing

( p , q ) out. The quantity p + can therefore be considered as a penalty (see [7]) for

fixing ( p , q ) out. The important thing to note is that the penalty p + can be computed

from an assignment solution without changing it any way. Consequently, the

penalties for the descendants of a node in the implicit enumeration approach can be

efficiently computed without altering the assignment solution for the parent node.

In the subtour elimination algorithms to be presented next, it becomes necessary

to “fix out” a basic cell ( p , q ) , i.e., to exclude the assignment ( p , 4). This can be

accomplished by applying the operator MC&, where M is a large positive number.

Similarly a cell ( p , q ) that was previously fixed out can be “freed”, i.e., its cost

restored to its true value, by applying the negative cell cost operator. A cell can

likewise be “fixed in” by applying MC,.



3. New LIFO implicit enumeration algorithms



The first algorithm (called TSP1) uses the Eastman-Shapiro subtour elimination

constraints with the modification suggested by Bellmore and Malone [ l , p. 3041 and

is a straightforward adaptation to the TSP of the implicit enumeration algorithm for

the zero-one integer programming problem. We first give a stepwise description of

algorithm TSP1:

Step 0. Initialize the node counter to zero and solve the AP. Initialize Z B = M

(ZB is the current upper bound on the minimal tour cost) and go to Step 1.

Step 1. Increase the node counter. If the current AP-solution corresponds to a

tour, update Z B and go to Step 4. Otherwise find a shortest subtour and determine

a penalty p + for each edge in this subtour (if the edge has been fixed in, take

p + = M, a large positive number, otherwise compute p + ) . Let ( p , q ) be any edge in

this subtour with smallest penalty p +. If Z + p + z=ZB, go to Step 4 (none of the

edges in the subtour can be fixed out without Z exceeding ZB). Otherwise go to

Step 2.



498



T.H.C. Smith, V. Sriniuasan, G.L. Thompson



Step 2 . Fix ( p , q ) out. If in the process of fixing out, Z + p + a ZB, go to Step 3.

Otherwise, after fixing ( p , q ) out, push (p, q ) on to the stack of fixed edges and go to

Step 1.

Step 3. Free ( p , q ) . If (9, p ) is currently fixed in, go to Step 4. Otherwise fix ( p , q )

in, push ( p , q ) on to the stack of fixed edges and go to Step 1.

Step 4. If the stack of fixed edges is empty, go to Step 6. If the edge (p, q ) on top

of the stack has been fixed out in Step 2, go to Step 3. Otherwise, go to Step 5.

Step 5. Pop a fixed edge from the stack and free it (if it is a fixed in edge, restore

the value of the corresponding assignment variable to one). Go to Step 4.

Step 6 . Stop. The tour corresponding to the current value of ZB is the optimal

tour.



In Step 1 of TSPl we select the edge (p, q ) to be fixed out as the edge in a shortest

subtour with the smallest penalty. Selecting a shortest subtour certainly minimizes

the number of penalty calculations while the heuristic of selecting the edge with the

smallest penalty is intuitively appealing (but not necessarily the best choice). We

tested this heuristic against that of selecting the edge with (i) the largest penalty

among edges in the subtour (excluding fixed in edges) and (ii) the largest associated

cost, on randomly generated asymmetric TSP’s. The smallest penalty choice

heuristic turned out to be three times as effective than (i) and (ii) on the average,

although it did not do uniformly better on all test problems.

Every pass through Step 1 of algorithm TSPl requires the search for a shortest

subtour and once an edge ( p , q ) in this subtour is selected, the subtour is discarded.

Later, when backtracking, we fix ( p , q ) in during Step 3 and go to Step 1 and again

find a shortest subtour. This subtour is very likely to be the same one we discarded

earlier and hence there is a waste of effort. An improvement of the algorithm TSPl

is therefore to save the shortest subtours found in Step 1 and utilize this information

in later stages of computation. We found the storage requirements to d o this were

not excessive, so that this idea was incorporated into the next algorithm.

The second algorithm, called TSP2, effectively partitions a subproblem into

mutually exclusive subproblems as in the scheme of Bellmore and Malone [1, p.

3041 except that the edges in the subtour to be eliminated are considered in order

of increasing penalties instead of the order in which they appear in the subtour.

Whereas the search tree generated by algorithm TSPl has the property that every

nonterminal node has exactly two descendants, the nonterminal nodes of the search

tree generated by algorithm TSP2 in general have more than two descendants. We

now give a stepwise description of Algorithm TSP2. In the description we make use

of the pointer S which points to the location where the Sth subtour is stored (i.e. at

any time during the computation S also gives the level in the search tree of the

current node).

Step 0. Same as in algorithm TSP1. In addition, set S = 0.

Step 1. Increase the node counter. If the current AP-solution corresponds to a

tour, update Z B and go to Step 4. Otherwise increase S, find and store a shortest



Computational performance of subtour elimination algorithms



499



subtour as the S t h subtour (together with a penalty for each edge in the subtour,

computed as in Step 1 of algorithm TSP1). Let ( p , q ) be any edge in this subtour

with smallest penalty p + . If Z + p + 3 ZB, decrease S and go to Step 4 (none of the

edges in the subtour can be fixed out without Z exceeding Z B ) . Otherwise go to

Step 2.

Step 2. Same as in algorithm TSP1.

Step 3. Free ( p , q). If all edges of the Sth subtour have been considered in Step 2 ,

decrease S and go to Step 4. Otherwise determine the smallest penalty p + stored

with an edge (e,f) in the S t h subtour which has not yet been considered in Step 2 . If

Z + p + < Z B , fix ( p , q ) in, push ( p , q ) on to the stack of fixed edges, set

( p , q ) = (e,f ) and go to Step 2. Otherwise decrease S and go to Step 4.

Step 4. Same as in algorithm TSP1.

Step 5. Same as in algorithm TSP1.

Step 6. Same as in algorithm TSP1.

The third algorithm, called algorithm TSP3, effectively partitions a subproblem

into mutually exclusive subproblems as in the scheme of Garfinkel [6]. A stepwise

description of the algorithm follows:

Step 0. Same as in algorithm TSP2.

Step 1. Increase the node counter. If the current AP-solution corresponds to a

tour, update Z B and go to Step 6. Otherwise increase S and store a shortest

subtour as the S t h subtour (together with a penalty for each edge in the subtour,

computed as in Step 2 of algorithm TSP1). Let ( p , q ) be the edge in this subtour with

smallest penalty p + . If Z + p + 2 ZB, go to Step 5. Otherwise go to Step 2 .

Step 2. Fix out all edges ( p , k ) with k a node in the Sth subtour. If in the process

of fixing out, Z + p t 2 ZB, go to Step 3. Otherwise, when all these edges have been

fixed out, go to Step 1.

Step 3. Free all fixed out (or partially fixed out) edges ( p , k ) with k a node in the

Sth subtour. If all edges in the S t h subtour have been considered in Step 2, go to

Step 4. Otherwise determine the smallest penalty p + stored with an edge ( e , f ) in

the Sth subtour which has not yet been considered in Step 2. If Z + p f < ZB, fix

out all edges ( p , k ) with k not a node in the S t h subtour, let p = e and go to Step 2.

Otherwise go to Step 4.

Step 4. Free all edges fixed out for the S t h subtour and go to Step 5.

Step 5. Decrease S. If S = 0, go to Step 7. Otherwise go to Step 6.

Step 6. Let ( p , k ) be the last edge fixed out. Go to Step 3 .

Step 7. Stop. The tour corresponding to the current value of Z B is the optimal

tour.

Note that the fixing out of edges in step 3 is completely optional and not required

for the convergence of the algorithm. If these edges are fixed out, the subproblems

formed from a given subproblem do not have any tours in common (see [6]). Most

of these edges will be nonbasic so that the fixing out process involves mostly cost



T.H.C. Smith, V. Sriniuasan, G.L. Thompson



500



changes. Only a few basis exchanges are needed for any edges that may be basic.

However, there remains the flexibility of fixing out only selected edges (for

example, only non-basic edges) or not fixing out of any of these edges.



4. Computational experience

Our major computational experience with the proposed algorithms is based on a

sample of 80 randomly generated asymmetric traveling salesman problems with

edge costs drawn from a discrete uniform distribution over the interval (1,1000).

The problem size n varies from 30 to 180 nodes in a stepsize of 10 and five problems

of each size were generated. All algorithms were coded in FORTRAN V and were

run using only the core memory (approximately 52,200 words) on the UNIVAC

1108 computer.

We report here only our computational experience with algorithms TSP2 and

TSP3 on these problems since algorithm TSPl generally performed worse than

either of these algorithms, as could be expected a priori.

In Table 1 we report, for each problem size, the average runtimes (in seconds) for

solving the initial assignment problem using the 1971 transportation code of

Table 1.

Summary of computational performance of algorithms TSP2 and TSP3

Average

Problem

size

n



time to

obtain

assignment

solution



30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

1 80



0.2

0.4

0.5

0.7

1.1

1.5

1.9

2.1

2.8

3.5

4.0

5.6

6.2

7.0

8.0

8.9



Note.



Algorithm TSP2

Average runtime

(including the

solution of the AP)

TSP2

TSP3

0.9

2.9

1.7

9.3

8.5

13.8

42.0

53.0

22.3

62.9

110.1

165.2

65.3

108.5

169.8

441.4



1.o

2.8

3.4

11.4

11.8

16.1

56.8

59.6



-



-



Average

runtime

Average time

Average quality

estimated by

to obtain

of first tour

regression

first tour

(% from optimum)

0.8

1.9

3.9

6.9

11.3

17.3

25.2

35.2

47.6

62.8

80.9

102.4

127.6

156.6

189.9

227.7



0.3

0.5

0.6

1.5

1.3

2.3

3.6

5.2

3.7

5.7

8.3

12.9

9.0

10.0

13.2

23.0



(1) All averages are computed over 5 problems each.

(2) All computational times are in seconds on the UNIVAC 1108.



3.7

4.0

0.8

4.1

0.5

1.0

2.7

3.8

1.3

1.5

2.0

4.2

1.1

1.1

1.3

3.1



Computational performance of subtour elimination algorithms



501



Srinivasan and Thompson [16] as well as the average runtime (in seconds including

the solution of the A P ) for algorithms TSP2 and TSP3. From the results for

n G 100, it is clear that algorithm TSP2 is more efficient than TSP3. For this reason,

only algorithm TSP2 was tested on problems with n > 100. We determined that the

function t ( n ) = 1.55 X

x n3.*fits the data with a coefficient of determination

(R’)of 0.927. The estimated runtimes obtained from this function are also given in

Table 1.

It has been suggested that implicit enumeration or branch-and-bound algorithms

can be used as approximate algorithms by terminating them as soon as a first

solution is obtained. In order to judge the merit of doing so with algorithm TSP2,

we also report in Table 1 the average runtime (in seconds) to obtain the first tour as

well as the quality of the first tour (expressed as the difference between the first tour

cost and the optimal tour cost as a percentage of the latter). Note that for all n the

first tour is, on an average, within 5% of the optimum and usually much closer.

We mentioned above that the fixing out of edges in step 3 of algorithm TSP3 is

not necessary for the convergence of the algorithm. Algorithm TSP3 was temporarily modified by eliminating the fixing out of these edges but average runtimes

increased significantly (the average runtimes for the 70 and 80 node problems were

respectively 24.3 and 25.5 seconds). Hence it must be concluded that the partitioning scheme introduced by Garfinkel [6] has a practical advantage over the original

branching scheme of Bellmore and Malone [l].

The largest asymmetric TSP’s solved so far appears to be two 80-node problems

solved by Bellmore and Malone [l]in an average time of 165.4 seconds on an IBM

360/65. Despite the fact that the IBM 360/65 is somewhat slower (takes about 10 to

50% longer time) compared to the UNIVAC 1108, the average time of 13.8 seconds

for TSP2 on the UNIVAC 1108, is still considerably faster than the

Bellmore-Malone [ 11 computational times. Svestka and Huckfeldt [ 181 solved

60-node problems on a UNIVAC 1108 in an average time of 80 seconds (vs. 9.3

seconds for algorithm TSP2 on a UNIVAC 1108). They also estimated the average

runtime for a 100 node problem as 27 minutes on the UNIVAC 1108 which is

considerably higher than that required for TSP2.

The computational performance of algorithm TSP2 was also compared with the

LIFO implicit enumeration algorithm in [ 121 for the asymmetric traveling salesman

problem using Held and Karp’s 1-arborescence relaxation. The 1-arborescence

approach reported in [12] took, on the average, about 7.4 and 87.7 seconds on the

UNIVAC 1108 for n = 30 and 60 respectively. Comparison of these numbers with

the results in Table 1 again reveals that TSP2 is computationally more efficient. For

the symmetric TSP, however, algorithm TSP2 is completely dominated by a LIFO

implicit enumeration approach with the Held-Karp 1-tree relaxation. See [ 131 for

details.

A more detailed breakdown of the computational results are presented in Table

2 (for TSP2 and TSP3 for n S 100) and in Table 3 (for TSP2 for n > 100). The

column headings of Tables 2 and 3 have the following interpretations:



T.H.C. Smith, V. Sriniuasan, G.L. Thompson



SO2



Table 2.

Computational characteristics of algorithms TSP2 and TSP3 for n



Problem



Gap



Pivots

TSP2 TSP3



Nodes

TSP2 TSP3



P30-1

P30-2

P30-3

P30-4

P30-5

P40-1

P40-2

P40-3

P40-4

P40-5

P50-1

P50-2

P50-3

P50-4

P50-5

P60-1

P60-2

P60-3

P60-4

P60-5

P70- 1

P70-2

P7G-3

P70-4

P70-5

P80- 1

P80-2

P80-3

P80-4

P80-5

P90- 1

P90-2

P90-3

P90-4

P90-5

P100-1

P100-2

P100-3

Plow

P100-5



2.48

3.25

1.31

1.62

4.06

2.52

2.94

8.64

1.13

0.24

0.20

0.37

1.65

2.56

3.28

1.22

2.42

0.77

1.64

0.55

1.16

1.52

2.20

1.79

3.05

0.36

0.47

1.34

1.23

1.27

0.64

0.87

1.17

0.99

0.84

1.83

0.72

0.54

0.93

0.95



187

174

402

175

250

127

352

1278

144

572

134

171

544

257

340

524

559

1611

2164

266

449

1863

309

622

1000

1005

819

1759

1357

832

543

841

5858

1822

3596

3080

1382

4341

603

3770



196

194

344

464

251

137

657

1136

177

514

134

173

1307

300

872

2935

1099

1029

1260

268

503

2676

310

883

1397

1078

885

2348

1597

994

570

831

7331

4282

4867

804

1741

8812

638

3990



11

6

24

14

17

5

16

58

7

44

2

3

31

7

11

13

23

65

92

3

8

94

3

21

34

36

25

43

25

29

3

5

226

37

140

90

35

144

8

160



12

6

24

14

17

5

16

51

7

25

2

3

54

7

11

112

22

37

32

3

8

103

3

21

34

33

25

47

27

27

3

5

217

102

139

8

36

248

8

88



Penalties

TSP2 TSP3

___

177

173

142

121

385

488

469

173

290

280

55

42

686

381

1459

1674

115

77

627

786

6

6

21

19

1768

613

236

192

908

350

4176

428

1147

605

1029

1611

1348

3279

27

29

312

260

3715

2630

31

30

878

62 1

1373

988

1154

1210

827

885

2636

1934

1658

1345

863

696

222

185

243

253

10860

9239

1835

5608

4990

6353

4294

254

1530

1914

6187 12877

193

226

6255

5232



C



100.



Maximum

subtours

stored

TSP2 TSP3

4

3

10

4

7

3

5

10

3

10

1

2

8

4

6

7

6

12

20

2

4

13

2

7

6

10

8

9

6

8

2

3

34

7

17

17

9

17

5

28



4

3

7

4

6

3

6

10

3

6

1

2

11

4

5

19

6

7

7

2

4

13

2

7



6

9

7

8

6

8

2

3

29

10

17

4

10

29



5

12



Runtime

(secs.)



TSP2



TSP3



0.7

0.5

1.6

0.7

1.o

0.4

1.9

7.7

0.6

3.7

0.4

0.5

3.9

1.5

2.3

4.1

4.9

12.5

24.2

1.0

3.4

22.9

1.1

6.4

8.9

13.3

9.1

21.4

15.6

9.6

4.4

6.3

108.1

24.6

66.4

61.4

23.3

94.3

4.8

81.1



0.7

0.6

1.4

1.3

1.0

0.5

2.8

6.9

0.8

3.0

0.3

0.4

10.5

1.6

4.2

30.4

7.0

8.5

10.2

0.9

4.1

34.6

1.1

7.7

11.3

13.2

10.1

28.4

17.6

11.3

4.3

5.9

129.8

67.8

76.0

5.9

29.8

182.4

4.8

75.0



Computational performance of subtour elimination algorithms



503



Table 3.

Computational characteristics of algorithm TSP2 for n > 100.



Problem



Gap



Pivots



Nodes



Penalties



Maximum

subtours

stored



Runtime

(secs.)



P110-1

P110-2

P110-3

P110-4

P110-5

P120-1

P 120-2

P 120-3

P 120-4

P120-5

P130-1

P130-2

P130-3

P130-4

P130-5

P140-1

P140-2

P140-3

P140-4

P140-5

P150-1

P 150-2

P150-3

P150-4

P150-5

P160-1

P 160-2

P 160-3

P160-4

P160-5

P170-1

P170-2

P170-3

P170-4

P170-5

P180-1

P180-2

P180-3

P180-4

P180-5



0.98

0.65

0.36

0.83

0.05

0.85

0.45

0.31

1.06

1.17

0.33

0.06

2.16

0.12

0.49

0.65

0.54

1.49

1.21

0.06

0.81

0.64

0.49

1.29

0.86

0.10

0.40

0.85

0.78

0.80

0.06

0.40

0.68

0.55

0.12

1.37

0.56

0.21

2.90

0.38



2948

1223

1141

1526

719

2754

2044

1526

1311

6046

7451

1985

5968

3107

2557

17.57

1568

11109

8772

2274

1491

4139

1597

2915

2788

3729

3683

3563

3250

3615

4133

3048

4311

4196

8080

12535

7115

13043

9292

7202



55

25

22

14

2

74

61

31

20

149

184

44

139

77

40

26

17

319

236

52

20

84

14

61

73

79

66

54

79

74

77

40

66

110

199

271

189

299

179

135



3605

1053

699

1162

9

3237

2396

1431

838

9336

11910

1804

8063

4264

2615

1067

895

19591

13684

2540

769

4902

680

2675

3151

4923

4056

3314

4363

4422

4393

2854

4119

6532

12577

19031

10614

21300

13900

9168



13

6

7

5

1

11

13

6

7

14

17

8

12

13

8

8

6

49

37

10

5

16

6

10

15

10

12

13

16

11

10

7

11

13

17

24

22

27

24

20



52.0

18.9

14.7

23.0

2.9

62.7

46.7

28.9

16.8

159.3

218.4

40.0

152.7

83.2

56.1

27.4

24.4

407.1

307.6

59.3

23.5

128.4

21.9

74.1

78.5

120.5

105.0

92.5

105.8

118.9

123.9

85.9

135.1

173.5

330.4

574.2

304.4

609.2

430.1

289.1



504



T.H.C. Smith, V. Sriniuasan, G.L. Thompson



The ith problem of size n is identified as Pn-i.

The difference between the optimal assignment cost and the optimal

tour cost as a percentage of the optimal tour cost.

The total number of basis exchanges.

Pivots :

The number of nodes in the search tree generated (i.e. the final value

Nodes :

of the node counter used in the algorithm descriptions).

The total number of times that p + or p were computed (either as a

Penalties :

penalty or in the process of fixing out or freeing a cell).

Maximum : The maximum number of subtours stored simultaneously (i.e. the

Subtours

maximum depth of a node in the search tree generated).

Stored

The total runtime in seconds on the UNIVAC 1108 including the time

Runtime :

for solving the A P but excluding time for problem generation.



Problem :

Gap :



~



From Tables 2 and 3 we find that the maximum number of subtours that had to

be stored €or a problem of size n was always less than n / 3 except for a 90 node

problem which had 34 maximum subtours and a 140 node problem which had 49

maximum subtours. Thus allowing for a storage of a maximum of about n / 2

subtours should suffice almost always.

In [2] Christofides considers asymmetric traveling salesman problems with

bivalent costs - i.e. each cost c,, i# j , can have only one of two values. H e

conjectured that this type of problem would be “difficult” for methods based on

subtour elimination and hence proposed and tested a graph-theoretical algorithm

for these special traveling salesman problems. In the testing of his algorithm (on a

C D C 6600) he made use of six problems ranging in size from 50 to 500 nodes. These

problems were randomly generated with an average of four costs per row being

zero and all nonzero costs having the value one (except for diagonal elements which

were M , as usual).

For each of the problem sizes 50, 100, 150 and 200 we generated five problems

(i.e. twenty problems altogether) with zero-one cost matrices (except for diagonal

elements) which have the same type of distribution of zeros as Christofides’

problems. We solved the problems with fewer than 200 nodes with both algorithms

TSPl and TSP2 and the five 200 node problems with algorithm TSPl only (because

of core limitations on the UNIVAC 1108 we are limited to 200-node problems for

algorithm TSPl and 180-node problems for algorithm TSP2).

The average runtimes (in seconds) for each problem size are reported in Table 4.

The last column of Table 4 contains the C D C 6600 runtime (in seconds) obtained by

Christofides on a problem of the given size. Since the C D C 6600 is generally

regarded as faster (takes about 10-50% less time) compared to the UNIVAC 1108,

algorithms TSPl and TSP2 can be regarded as more efficient than the algorithm in

[ 2 ] .An interesting observation was that for all the problems of this type which were

solved, the optimal assignment cost equalled the optimal tour cost (i.e., a n optimal

A P solution is also optimal to the TSP).



505



Computational performance of subtour elimination algorithms

Table 4.

Computational comparisons for bivalent cost asymmetric traveling salesman problems.

Problem

size

n



50

100

150

200



Average runtime"

(UNIVAC 1108 secs.)

TSPl

TSP2



0.5

1.4

5.4

6.4



Christofides' [2]

runtime

(CDC 6600 secs.)



0.6

1.5

5.4



9.5

15.9



-



12.8



-



Average based on 5 problems each.



5. Conclusion

We have proposed new algorithms for the asymmetric traveling salesman

problem and presented extensive computational experience with these algorithms.

The results show that our algorithms are:

(i) more efficient than earlier algorithms and

(ii) capable of solving problems of more than twice the size previously solved.

In view of the ongoing research on transportation .algorithms and the improvements

in computer performance, it is likely that the proposed algorithms will be able to

solve much larger traveling salesman problems in the near future.



References

[l] M. Bellmore and J.C. Malone, Pathology of traveling salesman subtour-elimination algorithms,

Operations Res. 19 (1971) 278-307.

[2] N. Christofides, Large scheduling problems with bivalent costs, Computer J. 16 (1973) 262-264.

[3] G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, Solution of a large scale traveling salesman

problem, Operations Res. 2 (1954) 393-410.

[4] G.B. Dantzig, D.R. Fulkerson and S.M. Johnson, On a linear programming, combinatorial

approach t o the traveling salesman problem, Operations Res. 7 (1959) 58-66.

[5] W.L. Eastman, Linear programming with pattern constraints, Unpublished Ph.D. Dissertation,

Harvard University (1958).

[6] R.S. Garfinkel, On partitioning the feasible set in a branch-and-bound algorithm for the

asymmetric traveling salesman problem, Operations Res. 21 (1973) 340-343.

[7] R.S. Garfinkel and G.L. Nemhauser, Integer Programming, (John Wiley, New York, 1972).

[8] K.H. Hansen and J. Krarup, Improvements of the Held-Karp algorithm for the symmetric

traveling salesman problem, Math. Programming 7 (1974) 87-96.

[9] M. Held and R.M. Karp, The traveling salesman problem and minimum spanning trees, Operations

Res. 18 (1970) 1138-1162.

[lo] M. Held and R.M. Karp, The traveling salesman problem and minimum spanning trees: Part 11,

Math. Programming 1 (1971) 6 2 5 .

[ I l l D.M. Shapiro, Algorithms for the solution of the optimal cost and bottleneck traveling salesman

problems, unpublished Sc. D. Thesis, Washington University, St. Louis, (1966).



506



T.H.C. Smith, V. Srinivasan, G.L. 7'hompson



[ 121 T.H.C. Smith, A LIFO implicit enumeration algorithm for the asymmetric traveling salesman

problem using a 1-arborescence relaxation, Management Science Research Report No. 380,

Graduate School of Industrial Administration, Carnegie-Mellon University (1975).

[13] T.H.C. Smith and G.L. Thompson, A LIFO implicit enumeration search algorithm for the

symmetric traveling salesman problem using Held and Karp's I-tree relaxation, Ann. Discrete

Math. 1 (1977) 479-493.

[14] V. Srinivasan and G.L. Thompson, An operator theory of parametric programming for the

transportation problem-I, Naval Res. Logistics Quarterly 19 (1972) 205-225.

[15] V. Srinivasan and G.L. Thompson, An operator theory of parametric programming for the

transportation problem-11, Naval Res. Logistics Quarterly 19 (1972) 227-252.

[16] V. Srinivasan and G.L. Thompson, Benefit-cost analysis of coding techniques for the primal

transportation algorithm, J. Assoc. Compuring Machinery 20 (1973) 194-213.

[17] V. Srinivasan and G.L. Thompson, Solving scheduling problems by applying cost operators to

assignment models, in: S.E. Elmaghraby (Ed.), Symp. Theory of Scheduling and its Applications,

(Springer, Berlin, 1973) 399-425.

[18] J. Svestka and V. Huckfeldt, Computational experience with an M-salesman traveling salesman

algorithm, Management Sci. 19 (1973) 790-799.



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