The General Traveling Salesman Problem, Version 5

TL;DR

This paper introduces a novel approach to the General Traveling Salesman Problem using prime number remainders and a chain-based iterative cycle reduction method to find shorter Hamiltonian cycles.

Contribution

It proposes a new matrix construction based on prime remainders and a chain process of 3-cycle permutations to improve cycle length in the GTSP.

Findings

The method reduces cycle lengths through iterative 3-cycle permutations.

Prime-based matrix encoding influences cycle optimization.

The approach offers a new heuristic for the GTSP.

Abstract

This paper is example 5 in chapter 5. Let H be an n-cycle. A permutation s is H-admissible if Hs = H' where H' is an n-cycle. Here we define a 19 X 19 matrix, M, in the following way: We obtain the remainders modulo 100 of each of the smallest 342 odd primes. we obtain the remainders modulo 100 of each of the primes. They are placed in M according to the original value of each prime. Thus their placement depends on the the original ordinal values of the primes according to size. We use this ordering to place the primes in M. Let H_0 be an initial 19 cycles arbitrarily chosen. We apply a sequence of up to [ln(n)+1] H_0 3-cycles to obtain a 19-cycle of smaller value than H_0, call the new 19-cycle H_1. We follow this procedure to obtain H_1. We call [ln(n)] + 1 a chain. We add up the values of the 19-cycles in each chain. This procedure continues until we cannot obtain a chain the sum of whose values is not negative. COMMENT. I've renamed the document "Yhe General Traveling Salesman Problem, Version 5". I preciously named it "The Traveling Salesman, Version 5". Although the algorithms work on the GTSP, I thought that more people would google it if it was named "The Traveling Salesman Problem." Rhar qas because my work is only available through arxiv.org,