an algorithm for the traveling salesman problem john d. c. little, katta g. murty, dura w. sweeney,...

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