A polynomial matrix processing heuristic algorithm for finding high quality feasible solutions for the TSP

Abstract

In this paper we present a simple heuristic algorithm to find high-quality, feasible solutions, for the traveling salesman problem (TSP). We hypothesize, that the quality of the initial solution provided by the proposed heuristic will improve the performance of the subsequent algorithm in terms of number of iterations required to reach a certain level TSP solution. The proposed heuristic does not attempt to compete against known TSP algorithms and heuristics, but instead, should be considered to serve as a “pre-processor”. The method provides a simple framework for testing new node selection and neighborhood rules. The cost matrix of origin and destination pairs is processed in a systematic way starting from a principal diagonal matrix element to find a feasible TSP tour. The matrix reduction, systematic moves in rows and columns, systematic elimination of rows and columns from further consideration, and the “reserved” column declaration, assure that the resulting sequence of nodes and edges forms a complete TSP tour. The process can be repeated from each principal diagonal element. The best TSP tour found can then be used, for example, as an input to another algorithm (e.g. the TABU search, simulated annealing, ant colony optimization, nearest neighbor, or another heuristic) to attempt to improve the tour further. It should be noted, that the proposed technique can also be used for testing of presence of cycles of a proposed solution provided by another algorithm. While the goal of the heuristic algorithm is to attempt to find the optimum tour, optimality cannot be guaranteed.