## 1 Introduction
On the other hand, the traditional ACO-based algorithms tend to require a considerable computer memory, which is necessary to keep an \(n\times n\) pheromone matrix. Typically, the time complexity of the selection of each next move using ACO is also costly. These drawbacks are addressed in some recent ACO-based algorithms in which, at each iteration of the calculation of the pheromone levels, the intermediate data are reduced storing only a limited number of the most promising tours in computer memory. With Partial ACO (PACO), only some part of a known good tour is altered. A PACO-based heuristic was proposed in [PERSON] (2018) and the experimental results for four problem instances from library Art Gallery were reported. Effective Strategies + ACO (ESACO) uses pheromone values directly in the 2-opt local search for the solution improvement and reduces the pheromone matrix, yielding linear space complexity (see, for example, [PERSON] (2018)). Parallel Cooperative Hybrid Algorithm ACO (PACO-3 Opt) uses a multi-colony of ants to prevent a possible stagnation (see, for example, [PERSON] et al. (2018)). In a very recent Restricted Pheromone Matrix Method (RPMM) ([PERSON] et al., 2018), the pheromone matrix is reduced with a linear memory complexity, resulting in an essentially lower memory consumption. Another recent successful ACO-based Dynamic Flying ACO (DFACO) heuristic was proposed by [PERSON] et al. (2018). Besides these ACO-based heuristics, we have compared our heuristics with other two meta-heuristics. One of them is a parallel algorithm based on the nearest neighborhood search suggested by [PERSON] et al. (2018), and the other one, proposedby [PERSON] et al. [12], is a Discrete Pigeon-Inspired Optimization (DPIO) metaheuristic. We have also implemented directly the Nearest Neighborhood (NN) algorithm for the comparison purposes (see Section 4 and Appendix A). In Table A1 in Appendix A, we give a summary of the above heuristics including the information on the type and the number of the instances for which these algorithms were tested and the number of the runs of each of these algorithms. Unlike these heuristics, the heuristic that we propose here is deterministic, in the sense that, for any input, it delivers the same solution each time it is invoked; hence, there is no need in the repeated runs of our algorithm. We have tested the performance of our algorithm on 218 benchmark problem instances (the number of the reported instances for the algorithms from Table A1 vary from 6 to 36). The relative error of our algorithm for the tested instances did not beat the earlier known best results; however, for some instances, our error was better than that of the above-mentioned algorithms (see Table 9 at the end of Section 3). The error percentage provided by our algorithm has varied from 0% to 17%, with an average relative error of 7.16%. The standard error deviation over all the tested instances was 0.03. In terms of the CPU time, our algorithm was faster than ones from Table A1 except for six instances from Art Gallery RPMM [9] and Partial-ACO [6], and for two instances from TSPLIB DPIO [12] were faster (see Table 10). Among all the comparisons we made, in about 92% of the cases, our algorithm has required less computational time. We have halted the execution of our algorithm for the two of the above-mentioned largest problem instances in 15 days, and for the next largest instance _ara238025_ with 238,025 cities our algorithm has halted in about 36 h. The average CPU time for the remained instances were 19.2 min. The standard CPU time deviation for these instances was 89.3 min (for all the instances, including the above-mentioned three largest ones, it was 2068.4 min). Our algorithm consumes very little computer memory. For the largest problem instance with 744,710 cities, it has used only about 50 MiB (mebibytes). The average memory usage for the remained 217 instances was 1.6 MiB (the average for all the instances including the above largest one was 1.88 MiB).