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Chapter 25. On Some Nonlinear Knapsack Problems

# Chapter 25. On Some Nonlinear Knapsack Problems

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Table 3’

n = 50

n = 70

n = m

n=80

Gap

Iterations

Nodes

Time

Gap

Iterations

Nodes

Time

Gap

Iterations

Nodes

Time

Gap

Iterations

Nodes

0.00

0.00

0.00

0.00

1.17

0.00

0.00

0.02

0.30

0.41

0.08

0.02

0.16

0.20

0.10

67

56

104

74

1863

1

1

2

1

70

1

1

7

39

21

27

27

21

19

15

2.8

2.4

4.3

3.1

79.5

3.5

2.5

8.6

46.7

27.2

37.2

38.9

30.4

25.2

18.8

0.00

0.53

0.00

0.00

0.26

0.00

0.24

0.51

0.10

0.00

0.09

0.10

0.00

0.00

0.06

418

858

65

76

1165

15

23

1

1

35

1

9

85

25

1

23

5

1

1

3

24.9

51.0

3.8

4.5

70.3

4.7

20.2

190.9

58.0

4.7

48.9

11.5

4.3

3.6

10.0

0.23

0.63

0.00

0.00

0.05

0.37

0.10

0.00

0.18

0.39

0.43

0.00

0.71

0.16

0.00

550

1417

82

108

219

800

191

66

2299

2687

1129

11

39

45.8

115.7

6.6

8.7

17.6

0.00

0.31

0.17

0.00

0.12

109

695

2530

125

361

1

11

59

84

60

207

1114

651

894

936

731

604

452

Average Runtime = 22.1

(I

= 0.01,

p = 0.1,

T = 0.9

80

341

3211

977

80

820

193

73

61

169

Average Runtime = 34.1

80

1104

555

117

1

1

3

22

3

1

67

63

29

1

25

15

1

1

5

Time

11.8

75.4

275.4

13.6

38.9

64.5

15.4

5.3

185.7

217.6

91.2

6.4

89.7

44.9

9.4

Average Runtime = 61.6

Average Runtime = 83.0

in all problems.

P

s

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

492

be expected to provide a good upper bound U, for the type of problem under

consideration, we took as upper bound 1.01 times the value of the lower bound at

the end of the initial ascent (if no tour is found, the algorithm has to be run again

with a higher upper bound). In the initial ascent we computed the stepsize as

t = A ( 0 . 5 M ) / ~ z , N (-d2)*

, (where M is the maximum lower bound obtained in the

d ,2)’. All

current ascent) while in a general ascent we took t = A(0.005 M ) / Z I E N ( ascents were started with the parameter K set equal to 2 , the threshold value.

Our computational experience with the above fifty problems are reported in

Table 3 where the column headings have the following interpretations:

Gap:

Iterations:

Nodes:

Time :

The difference between the optimal tour length and the lower bound

at the end of the initial ascent as a percentage of the optimal

tourlength.

The total number of ascent iterations.

The total number of subproblems generated in steps 2(b), 2(c) and

3(d) of IE.

The total runtime in seconds on the UNIVAC 1108.

A 100 node problem was also solved and took 13.6 minutes on the UNIVAC 1108,

generating 95 nodes and requiring 5014 ascent iterations.

For the reasons stated above it is extremely difficult to compare the IE-algorithm

with that of Hansen and Krarup. However, there does exist the possibility of

improving the IE-algorithm further by making use of efficient sorting techniques

and Kruskal’s algorithm for finding a minimal spanning tree (see ) as done by

Hansen and Krarup.

6. Conclusions

Our computational results indicate that the IE-algorithm is considerably faster

than the HKI-algorithm. Since the major computational effort in the IE-algorithm

is spent on Step 1 (the ascent method) in order to find good lower bounds on

subproblems, an increase in the efficiency of the algorithm can be obtained by

speeding up the ascent method. We are currently considering techniques for doing

the latter.

References

[l] E. Balas, A n additive algorithm for solving linear programs with zero-one variables, Operations

Res. 13 (1965) 517-546.

 N. Cristofides, T h e shortest hamiltonian chain of a graph, S I A M J . Appl. Math. 19 (1970) 689-696.

 E.W. Dijkstra, A note on two problems in connection with graphs, Num. Math. 1 (1959) 269-271.

[3a] L.R. Ford, Jr. and D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, NJ,

1962).

A LIFO implicit enumeration search algorithm

493

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

 A.M. Geoffrion, Integer programming by implicit enumeration and Balas’ method, SIAM Reo. 7

(1967) 178-190.

 F. Glover, A multiphase-dual algorithm for the zero-one integer programming problem, Operations

Res. 13 (1965) 879-919.

 F. Glover, D. Klingman and D. Karney, The augmented predecessor index method for locating

stepping stone paths and assigning dual prices in distribution problems, Transportation Sci. 6 (1972)

171-1 80.

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

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

 M. Held and R.M. Karp, A dynamic programming approach to sequencing problems, SIAM 10

(1962) 196210.

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

Res. 18 (1970) 1138-1162.

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

Programming 1 (1971) 6 2 5 .

 M. Held, P. Wolfe and H.P. Crowder, Validation of subgradient optimization, Math. Programming

6 (1974) 62-88.

 E. Johnson, Network and basic solutions, Operations Res. 14 (1960) 619-623.

1141 R.L. Karg and G.L. Thompson, A heuristic approach to solving traveling salesman problems,

Management Sci. 10 (1964) 225-248.

 J. Kershenbaum and R . Van Slyke, Computing minimum trees, Proc. A C M Annual Conference

(1972), 518-527.

 J.B. Kruskal, O n the shortest spanning subtree of a graph and the traveling salesman problem,

Proc. A m . Math. SOC. 2 (1956) 48-50.

 J.D.C. Little, et al., An algorithm for the traveling salesman problem, Operations Res. 11 (1963)

972-989.

 A.K. Obruca, Spanning tree manipulation and the traveling salesman problem, Computer J. 10

(1968) 374-377.

1191 R.C. Prim, Shortest connection networks and some generalizations, Bell System Tech. J., X (1957)

1389- 1401.

 T.C. Raymond, Heuristic algorithm for the traveling salesman problem, IBM J. Res. Deu., 13 (1969)

400-407.

 P. Rosenstiehl, L’arbre minimum d’un graphe, in: P. Rosenstiehl, ed., Theory of Graphs (Gordon

and Breach, New York, 1967).

 V. Srinivasan and G.L. Thompson, Accelerated algorithms for labeling and relabeling of trees, with

applications to distribution problems, J. Assoc. Computing Machinery 19 (1972) 712-726.

1231 G.L. Thompson, The stopped simplex method: I. Basic theory for mixed integer programming,

integer programming, Reuue Francaise D e Recherche Operationelle 8 (1964) 159-182.

 G.L. Thompson, Pivotal operations for solving optimal graph problems, working paper.

Annals of Discrete Mathematics 1 (1977) 495-506

@ North-Holland Publishing Company

COMPUTATIONAL PERFORMANCE OF THREE SUBTOUR

ELIMINATION ALGORITHMS FOR SOLVING ASYMMETRIC

TRAVELING SALESMAN PROBLEMS*

T.H.C. SMITH

Department of Statistics, Rand Afrikaans University, Johannesburg, R.S.A.

V. SRINIVASAN

G.L. THOMPSON

Graduate School of Industrial Administration, Carnegie -Mellon Universify, Pittsburgh, P A 15213,

U.S.A.

In this paper we develop and computationally test three implicit enumeration algorithms for

solving the asymmetric traveling salesman problem. All three algorithms use the assignment

problem relaxation of the traveling salesman problem with subtour elimination similar to the

previous approaches by Eastman, Shapiro and Bellmore and Malone. The present algorithms,

however, differ from the previous approaches in two important respects:

(i) lower bounds on the objective function for the descendants of a node in the implicit

enumeration tree are computed without altering the assignment solution corresponding to the

parent node - this is accomplished using a result based on “cost operators”,

(ii) a LIFO (Last In, First Out) depth first branching strategy is used which considerably

reduces the storage requirements for the implicit enumeration approach. The three algorithms

differ from each other in the details of implementing the implicit enumeration approach and in

terms of the type of constraint used for eliminating subtours. Computational experience with

randomly generated test problems indicates that the present algorithms are more efficient and can

solve larger problems compared to (i) previous subtour elimination algorithms and (ii) the

1-arborescence approach of Held and Karp (as implemented by T.H.C. Smith) for the asymmetric

traveling salesman problem. Computational experience is reported for up to 180 node problems

with costs (distances) in the interval (1,1000)and up to 200 node problems with bivalent costs.

1. Introduction

Excluding the algorithms of this paper, the state-of-the-art algorithms for the

asymmetric traveling salesman problem appears to be that of [ll] and more

recently [l],both of which use the linear assignment problem as a relaxation (with

subtour elimination) in a branch-and-bound algorithm. In the case of the symmetric

* This report was prepared as part of the activities of the Management Science Research Group,

Carnegie-Mellon University, under Contract N00014-75-C-0621 NR 047-048 with the U.S. Office of

Naval Research.

A considerably more detailed version of this paper is available (Management Sciences Research

Report No. 369), and can be obtained by writing to the third author.

495

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  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  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 .

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 , recently proposed and tested computationally in .

2. Subtour elimination using cost operators

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

Eastman , 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 ) 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  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 ) 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 ) 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 . 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 ). 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  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  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  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: ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Chapter 25. On Some Nonlinear Knapsack Problems

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