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An Algorithm for theTraveling SalesmanProblem

 John D. C. Little, Katta G. Murty,Dura W. Sweeney, and Caroline

Karel

1963

Speaker: Huang Cheng-Kang

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What’s TSP? A salesman, starting in one city, wishes

to visit each of n – 1 other cities once and

only once and return to the start.

In what order should he visit the cities tominimize the total distance traveled?

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 The Algorithm  The basic method:

Branch – break up the set of all tours

Bound – calculate a lower bound

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Notation (1/2)  The entry in row i and column j of the

matrix is the cost for going from city i to

city j. Let

A tour, t , can be represented as a set of n 

ordered city pairs, e.g.,

== )],([ jicC  cost matrix

)],)(,(),)(,[( 113221 iiiiiiiit  nnn−=

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Notation (2/2)  The cost of a tour, t, under a matrix, C, is the sum

of the matrix elements picked out by t and will bedenoted by :

Also, let

nodes of the tree;

a lower bound on the cost of the tours of   X ,i.e., for t a tour of  X ;

the cost of the best tour found so far.

)(t  z 

∑= ),()( ji

t  z ),( jicint 

=)( X w

=Y Y  X  ,,

)()( X wt  z  ≥

=0 z 

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Lower Bounds  The useful concept in constructing

lower bounds will be that of reduction.

t = [(1,2) (2,3) (3,4) (4,1)] z = 3+6+6+9 = 24

Reduce

Reduce Concept:

at least one zero in each row and colu

 z = 16+(3+8) = 24

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Branching

all tours

 ji, ji,

l k ,l k ,

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4

1

0 1

6

6 the sum of the smallest

element in row i and column j

:

∞

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∞

 下限 = 170

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all tours

3,43,4

16

1722

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4

01

0 0∞

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all tours

3,43,4

16

1722

2,12,11721

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all tours

3,43,4

16

1722

2,12,11721

4,24,2

17

1,317

t = [(1,2) (2,3) (3,4) (4,1)] z = 3+6+6+9 = 24