An Algorithm for the Traveling Salesman Problem

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

1963

Speaker: Huang Cheng-Kang

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 to minimize the total distance traveled?

The Algorithm The basic method:

Branch – break up the set of all tours Bound – calculate a lower bound

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

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 be denoted 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.

)(tz

),()( jitz ),( jicin t

)(XwYYX ,,

)()( Xwtz 0z

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 column

z = 16+(3+8) = 24

Branching

all tours

ji,ji,

lk,lk,

4

1

0 1

6

6 the sum of the smallest element in row i and column j

:

下限 = 170

all tours

3,43,4

16

1722

4

01

0 0

all tours

3,43,4

16

1722

2,12,11721

all tours

3,43,4

16

1722

2,12,11721

4,24,217

無解

1,317t = [(1,2) (2,3) (3,4) (4,1)]z = 3+6+6+9 = 24