At the beginning it was pure coincidence, at the end everything is allways totally easy.

The story begins some twenty years ago. The author, a trained programmer, first met with the traveling salesman problem (TSP) back in 1985. The Fachschullehrbuch Mathematik für Wirtschaftswissenschaften, Verlag die Wirtschaft, Berlin (DDR),1983, described the TSP in a straight forward and at first side easy way so that a quick solution was found. It was not very long that the author realized that an increasing number of vertices indicated a no longer optimum solution (i.e. a complete enumeration of every single partial solution). Hence he wanted complete reconsideration of all the problems connected with this optimum solution.

After the usual trial and error solutions the more or less disappointed author started again "with the beginning".  As a programmer he was used to living with numbers, and was therefore troubled by the notion that any round trip resulted with the number two for every given vertex, meaning that any given edge might have this vertex either as starting or as an end point. The question arose, what would be the solution, if every vertex existed only once. Further: what would result with a round trip of six vertices only? He commenced listing up the possibilities, until he realized that there were only a few basic structures. That was the decisive moment. Any further statements - right up to the present moment - are nothing but consequences of these basic ideas.

The author stated that for 5! = 120 possible round trips with six vertices there were only 15 basic structures (partial round trips). The remaining parts were of exactly the same structure. That is any partial round trip may be completed to a total round trip, if the eight suitable complements were found. The algorithm was obvious: Take 1 x 3 x 5 x  ... instead of 1 x 2 x 3 x 4 x 5 x 6 x  ... . The remaining product 2 x 4 x 6 x ... indicates the exact number of the complementary partial round trips. Even this way of separation was seemingly not yet known (29.3.1985).

In 1998 the author discovered the general formula for the power of the partial graphs (he used the equivalence relation between graphs and round trips). By that formula the mathematic explanation was given as well: The faculty expression in the denominator gives the permutation of the free edges, the power of 2 in the denominator indicates the variation of the free edges under symmetry. On behalf of the mode of separation and the subsequent recombination it looked all like the working of a zip/zipper; therefore the new method was termed "ZIP-method" by the author. 

Is the "ZIP-method" known or not known in the scientific community?

Of course, a definit answer to this question may not be given. There are two reasons, however, may be offered that this method is not yet generally known:

1.
The structure of separation is surprisingly easy, and, in any case, extends the upper limit of the optimally calculable round trips. Hence, one can allways assume that the "ZIP-method" would be published if it were known. None of the many surveys for any possible hint of book or paper dealing with TSP produced a positiv result of the "ZIP-method". Even learned men like Grötschel and Padberg (Grötschel M, Padberg M „Die optimierte Odyssee“ in: Spektrum der Wissenschaft 4/99, S.80 ff.) state in 1999 that the TSP based on combinatorics could not offer optimum for 25 vertices. Therefore the "ZIP-method" seems to be realy unkown.

2.
Above all, there is an obvious explanation, why the "ZIP-method" had not yet been discovered:
Everyone, deeling with TSP, is allmost immediatly caught by a geographically based idea of round trips. Allmost at once a notation of a serial grouping of distances: The end of one distance is the starting point of the following one. Therefore MÜLLER-MERBACH also speaks of an "linear ordering problem ". It is this geographical imagination that (reably) blockes the way to the present solution. Only the taking into account of the mere numbers, i.e. the using of algebra, brought the author to this solution by chance.

(18.09.2004)