The Traveling Salesman Problem
 

A traveling salesman wishes to make a number of sales calls in different
cities. Naturally he wants to spend as little time actually traveling as
possible (he is not paid for travel time).

He sets up the various cities he wants to visit and looks up the travel time
from each one to every other one. He writes a simple computer program to figure
out all of the possible routes and compare the total travel time in each route,
in order to find the fastest possible route. (The problem could also be worked
for minimum distance).

The program is written for any arbitrary number of cities. But the salesman
discovers that the program appears not to work for more than a certain number
of cities. Explain what the limit is and how it can be overcome.

73 de Jim, N2EY

This looks like a problem from Discrete Mathematics. To me it appears to be
a specific problem in finding what is called the minimum "Spanning Tree" for
a graph. For your problem the graph vertices would be the cities visited and
the distance, or what ever you want to minimize, assigned to the edges
connecting the vertices together.

Two known algorithms are used to find a set of edges connecting all the
vertices together that will minimize the item assigned to the edges as a
weight factor. The two algorithms are "Prim's" and Kruskal's". Their work
was done during the 1950's. At most (N-1) edges, where N is the total number
of edges in the graph, are selected during the course of the algorithm.
Nether algorithm has any limit on the size of the graph, vertices or number
of edges, that I can find.

(Robert Clay Prim born 1921 Swee****er Texas)
(Joseph Bernard Kruskal born 1928 New York city)

The book I have as a reference is:

Discrete Mathematics And Its Applications
(Third Edition 1995)
Published by McGraw Hill
By Kenneth H. Rosen
ISBN 0-07-053965-0

Look at pages 593 (section 8.6) to 598.

There are some older methods that use what is called "depth-first searches",
or "back tracking", and the other is called "breadth-first search". These
are covered under (section 8.5) pages 583 to 588.

The problem with writing code for graph transversals is two fold. The first
is keeping track of where you have been, and second is not getting stuck in
what is called a "cycle" or a loop in the graph. The way I've seen this done
is using recursive code while building a tree structure of the nodes
"visited", where each tree entry is for a graph node. The code starts at a
beginning node, or current node, then follows every path from the current
node to the next node if the path is not already listed in the tree entry
for the current node. The new path is entered in the path list for the
current node. If the path is already listed then it has already been
explored and should be skipped. If the next node is not in the tree then
enter it and make it the current node, other wise keep looking for another
path from the current node location to another node. If no more paths exists
from the current node then return to the previous node, making it the
current node, while marking the path as blocked. When no more paths exist to
explore at the current node then return to the previous node and explore the
next path not yet transversed. This procedure continues until either a path
to the destination node is found, or all nodes and paths have been explored.
Then doing a tree transversal "in order" looking for paths not marked as
blocked will generate a list of nodes and paths between nodes spanning from
the begining node to the destination node.

Graph theory is interesting and has a lot of practical applications in the
real world.

--
Leland C. Scott
KC8LDO

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"N2EY" wrote in message
...
A traveling salesman wishes to make a number of sales calls in different
cities. Naturally he wants to spend as little time actually traveling as
possible (he is not paid for travel time).

He sets up the various cities he wants to visit and looks up the travel
time
from each one to every other one. He writes a simple computer program to
figure
out all of the possible routes and compare the total travel time in each
route,
in order to find the fastest possible route. (The problem could also be
worked
for minimum distance).

The program is written for any arbitrary number of cities. But the
salesman
discovers that the program appears not to work for more than a certain
number
of cities. Explain what the limit is and how it can be overcome.

73 de Jim, N2EY