## My OPE & the Euclicidean TSP

A little while back I posted for a while on a creative algorithm I 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:

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/