Local search for the Traveling Salesman Problem: A comparative study

The Traveling Salesman Problem (TSP) is one of the most well-studied combinatorial optimization problems. Best heuristics for solving the TSP known today are Lin-Kernighan (LK) local search methods. Recently, Multi-Neighborhood Search (MNS) has been proposed and was demonstrated to outperform Variable Neighborhood Search based methods on the TSP. While LK performs a variable k-opt based search operation, MNS is able to carry out multiple 2-, 3-, or 4-opt moves at once, which are discovered by a highly efficient scan of the current solution. Apart from LK and MNS, many other modern heuristics for TSPs can be found in the relevant literature. However, existing studies rarely use robust statistics for the heuristic algorithms in comparison, let alone investigate their progress over time. This leads to flawed comparisons of simple end-of-run statistics and inappropriate or even incorrect conclusions. In this paper, we thoroughly compare LK and MNS as well as their hybrid versions with Evolutionary Algorithms (EAs) and Population-based Ant Colony Optimization (PACO). This work, to the best of our knowledge, is the first statistically sound comparison of the two efficient heuristics as well as their hybrids with EAs and PACO over time based on a large-scale experimental study. We not only show that hybrid PACO-MNS and PACO-LK are both very efficient, but also find that the full runtime behavior comparison provides deeper and clearer insights whereas a focus of final results could indeed have led to a deceitful conclusion.