The General Traveling Salesman Problem, Version 5

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,

💡 Research Summary

The manuscript titled “The General Traveling Salesman Problem, Version 5” attempts to introduce a novel framework for solving instances of the General Traveling Salesman Problem (GTSP), focusing on a specific 19‑node complete graph whose cost matrix is constructed from the residues modulo 100 of the first 342 odd primes. The author’s approach is built on two main ideas: (1) constructing a “relative upper bound” using a minimum‑cost perfect matching σ, and (2) iteratively improving a tour through H‑admissible 3‑cycle transformations.

In the first stage, the author defines σ as a set of n/2 disjoint 2‑cycles (a perfect matching) that yields the smallest possible sum of edge costs in the symmetric cost matrix M. By permuting the columns of M according to σ and subtracting each diagonal entry from its row, the author obtains a derived matrix denoted σM⁻⁻. On σM⁻⁻ a modified Floyd‑Warshall (F‑W) algorithm is applied. The modification consists of forbidding any intermediate arc that would be symmetric to an arc already present in the current derangement, thereby ensuring that only “acceptable paths” and “2‑circuit paths” are generated. Acceptable paths are those that never contain two vertices belonging to the same 2‑cycle of σ, while 2‑circuit paths allow a limited amount of such overlap under specific interleaving conditions. The author claims that any tour whose total cost is less than the cost of the initial upper bound can be expressed as a patchwork of these acceptable and 2‑circuit cycles.

The second stage introduces H‑admissible 3‑cycle operations. Starting from an arbitrary initial 19‑cycle H₀, the algorithm selects up to ⌈ln n⌉ + 1 three‑vertex permutations (3‑cycles) that strictly reduce the tour’s total cost, producing a new tour H₁. This process is repeated, forming a “chain” of tours; the sum of the costs of tours within a chain is accumulated, and the chain is extended as long as the cumulative sum remains non‑negative. When no further cost‑reducing 3‑cycle can be found, the procedure terminates.

The paper presents a series of theorems (1.1 through 1.9) intended to provide theoretical justification. Theorem 1.1 guarantees the existence of a “determining vertex” in any cycle such that the partial sums of arc weights from that vertex are non‑positive, a property used to argue that a cost‑reducing 3‑cycle can always be found. Theorem 1.2 restates the classic Floyd‑Warshall result, while Theorem 1.3 claims that if the matrix contains a negative cycle, the modified F‑W algorithm can isolate it using fewer columns than a generic non‑simple path would require. Subsequent theorems connect acceptable cycles in σM⁻⁻ to unique perfect matchings (Theorem 1.9) and demonstrate how alternating edges of a tour can be partitioned into a lower‑cost matching (Theorem 1.6).

Despite the ambitious scope, the manuscript suffers from several critical deficiencies. The notation is inconsistent; symbols such as σM⁻⁻, 1σM⁻⁻, and σ⁻¹M appear without clear definitions, making it difficult to follow the matrix transformations. The definitions of “acceptable path” and “2‑circuit path” are vague, and the algorithmic steps for extracting and patching cycles are not presented in pseudocode or flow‑chart form, hindering reproducibility. The complexity analysis is absent—while the author mentions that at most ⌈ln n⌉ + 1 3‑cycles are applied per iteration, there is no discussion of how many iterations are expected, nor of the overall polynomial versus exponential behavior. Moreover, the paper provides no empirical evaluation: there are no benchmark instances, runtime measurements, or comparisons with established heuristics such as 2‑opt, Lin‑Kernighan, or modern meta‑heuristics. Consequently, the claim that the method offers a practical advantage remains unsubstantiated.

The proofs of the theorems are largely hand‑wavy. Many rely on induction but omit crucial base cases or fail to justify key inductive steps, especially when handling alternating sign patterns in arc weights (Theorem 1.1). Theorem 1.3’s assertion about column usage in the modified F‑W algorithm is not backed by a rigorous argument about the algorithm’s internal data structures. The relationship between the derived matrix σM⁻⁻ and the original cost matrix M is not formally proved, leaving open the possibility that the transformation could introduce artifacts that invalidate the subsequent reasoning.

In summary, the manuscript introduces an interesting conceptual blend of perfect‑matching based decomposition and Floyd‑Warshall‑style path refinement for GTSP, but the presentation lacks the mathematical rigor, algorithmic clarity, and experimental validation required for a credible contribution to the field. To become publishable, the authors would need to (1) standardize notation and provide precise definitions, (2) supply detailed pseudocode for each algorithmic component, (3) conduct a thorough complexity analysis, (4) benchmark the method against standard TSP/GTSP solvers on a variety of instance sizes, and (5) strengthen the proofs of the central theorems. Until such revisions are made, the paper remains of limited interest to researchers and practitioners working on combinatorial optimization.