J

#### JSH

came up with for tracing out a path through nodes, and discussions got

bogged down on the issue of whether or not it solved the TSP, where

the consensus of several members was that it did not. But I have made

a project for the full algorithm at Google Code for coding in java and

while the welcome mat for coders has been put out, I have no

responses, so I have decided to talk more about the algorithm here

with the Euclidean TSP as I realized it'd be simpler to explain.

I will not give the full detailed algorithm here as I wish to simply

explain, but the gist of it is that you have TWO travelers who start

from the same node, every node is connected to every other node, and

the weight is the distance between nodes as it's the Euclidean

Traveling Salesman Problem being considered, and the two travelers

choose nodes such that they stay as physically close together as

possible where they can't choose the same node.

The weird thing in considering that as a solution is wondering how

local choices can have a global impact as playing with any TSP problem

for any length of time can, I'm sure, lead to the belief that the main

issue has to do with unknowns far away from the initial nodes, while

my idea says that local choices from BOTH sides of the path solve the

problem, so to help understand how local solves global, consider two

other travelers not using the algorithm.

To make it easier to imagine let's say the nodes are cities, and you

have two teams, where both teams are couples, and they all start from

the same city, but as they travel through all nodes--say, going

through European cities--they avoid again going to the same city, or

to a city their couple has already visited, but the first couple tries

to stay as close together physically as possible in their choices

while the other couple doesn't care, and makes different choices.

What happens after iteration 1?

Well the first couple has moved from the starting city to two other

different cities, choosing them such that their physical distance

apart is the LEAST possible given all possible city choices, while the

other couple has gone to different cities for some other reason, so

what do we now know?

We know that the second couple is further away from each other as they

traveled FARTHER than the first and MUST eventually make up that

distance, as eventually they come back together, so we already know

that the second couple has already traveled further and will have to

travel still further to make up the distance than the first. It's

like a double whammy. They traveled to more distant cities, and are

farther apart so will have a greater distance to travel in coming back

together down the line.

You may say, but what about the second choice, and the next and the

next?

Well, in each case the first couple remains as close as possible so

the second couple gets further behind, but can actually catch up as

the first couple can kind of bounce off each other if they're

traversing through very close cities until they're forced apart by

running through all of those so they have to get further apart as they

go to unvisited cities, so here is where the other couple can start

catching up.

Eventually each couple comes to a point where they're each at the last

two cities, so they can just pick one at which to meet, or there is

only one city left in the middle and they both move forward to meet

there, and tracing out the two routes you have just two routes along

which you can imagine a SINGLE traveler.

So at the end of the exercise you can collapse out the second traveler

and have a route for a single traveler in each case.

My hope is that pondering that problem and how each local choice leads

to a global result: distance apart, will help understanding of how

this algorithm works, and why it works.

Maybe the simplest thing for those of you who actually play with TSP

problems is to trace out a route for a Euclidean TSP, using two

travelers, where one starts at the end and works back to one starting

from the beginning and working forward and check the distance between

them at any point, versus two travelers using a non-optimal path.

My problem solving methods often involve using additional variables--

more degrees of freedom--which just help with solving the problem but

collapse out from the final solution and here using two travelers

allows a handle to be placed on the optimal path, which handles the

global problem piecewise with local decisions from BOTH ends.

I generalized the full algorithm to handle the TSP in general, where

you may not necessarily have distance information, and then I

generalized to situations where all nodes are not connected to every

other node, and got the full algorithm for what I call the optimal

path engine, or the OPE, which is waiting to be designed and coded.

The project space is optimalpathengine at Google Code. There is also

a newsgroup:

http://groups.google.com/group/optimalpathengine

Where you can discuss the idea including criticizing it if you like.

I'll only manage to the extent that I keep out flaming or any other

kind of deliberately disruptive behavior, so if you post there

disagreement with the idea, don't worry I won't get rid of it, though

if you're looking to simply sabotage the project with criticism, no

need to bother as so far nothing is happening anyway, which is why I'm

posting.

As a sidenote, for those interested in more in theory, if you look for

paths that are not round trip, so you're going to have a starting

node, and a different ending node, the algorithm behaves rather

interestingly in that if you start and finish at opposite ends then

the algorithm works in reverse in that you have the travelers pick in

a way that maximizes the distance between them, as otherwise they will

take the LONGEST path. Also, you can get the longest path with the

original by having them pick to maximize the distance between them.

Oh yeah, in closing, if this algorithm does work to pick the shortest

path then it proves that P=NP which is worth mentioning because the

solution then explains why "hard" problems are hard as they require

additional degrees of freedom not evident in the final solution e.g.

the second traveler of the algorithm and the distance between the two

travelers.

These additional degrees of freedom give the range necessary for

solving NP hard problems, but are invisible to people searching for

solutions unless they figure out an angle, so they can work for as

long as they won't and find various techniques that don't provide a

general solution, and yes, I have used additional variables in other

areas and I did go to TSP because I had this insight about this

problem solving approach and the TSP was the natural thing to

consider. The approach I use was born December 1999 out of attempts

at proving Fermat's Last Theorem. I'd exhausted very way I knew of

paying with x^p + y^p = z^p, so I thought to myself, wouldn't it be

neat if I had more degrees of freedom? So I've used the approach now

for over 8 years with amazing successes that are the subject of

controversy.

Another example of a problem where I used an additional degree of

freedom is my prime counting function, which is worth mentioning again

because of the reception it receives, as in chilly. There I found a

much simpler way to count prime numbers than is currently taught where

I have a P(x,y) function (fully mathematicized, but a P(x,n) in sieve

form), versus the pi(x) function of traditional mathematics.

It has been six years since that innovation. I have little

expectation that a solution proving P=NP would be rapidly picked up--

against the intuition or gut feelings of many of you I'm sure--but

fully expect MASSIVE resistance against the solution without any

objections being given that show the idea is actually flawed!!!

Amazingly enough.

(Consider that I actually had some of my research published in a

mathematical journal once. Readers on the sci.math newsgroup found

out about it, some conspired in posts an email campaign against the

paper. The editors just yanked my paper after that email campaign,

after publication, as it was an electronic journal, so they just left

a gap! They managed one more edition and then the entire math journal

shut-down. Its hosting university, Cameron University, part of the

Oklahoma state university system, removed ALL MENTION of the journal

from its website. That math journal had been around for 9 years. The

mathematical paper published in it over that timeframe might have been

lost except EMIS maintained the archives. Don't believe that amazing

story? See for yourself:

http://www.emis.de/journals/SWJPAM/

Link to edition that HAD my paper:

http://www.emis.de/journals/SWJPAM/vol2-03.html

An entire mathematical journal died quietly over a controversial paper

accepted from a supposed "crackpot" and the world just kept on going

like nothing happened. No big news story. No intrepid reporters from

ANY of the world's press that bothered to care--even though I tried to

bug them about it!!!

Revolutionary results run into the problem of defense against the

truth.)

But I could be wrong, so here's this post to see. Obviously if you

study this idea and see viability or prove it (remember you can trace

out actual Euclidean TSP solutions) then you can sign on and help

design and code the OPE. Even if you think I'm right, make no

mistake, it could take YEARS or even decades before the world accepts

the truth as that's how it really works, unlike fantasies some may

have from stories, legends or Hollywood movies. The fantasy world is

not the reality. Reality is a slog through the mud, and massive

resistance against the truth, and lots of people maybe willing to say

really mean things to you for a period of years.

There is no instant on, I like to say. So you can face ridicule, or

mostly being totally ignored for years no matter what you can prove,

or demonstrate even with a program. So only those who can move

forward without that social stuff like approval and accolades need

even consider signing on.

James Harris