A Multilevel Lin-Kernighan-Helsgaun Algorithm for the Travelling Salesman Problem

The multilevel paradigm has recently been applied to the travelling salesman problem with considerable success. The resulting algorithm progressively coarsens the problem, initialises a tour and then employs a local search algorithm to refine the solution on each of the coarsened problems in reverse order. In the original version the chained Lin-Kernighan (CLK) scheme was used for the refinement. However, a new and highly effective Lin-Kernighan variant (LKH) has recently been developed by Helsgaun. Here then we report on the modifications required to develop a multilevel LKH algorithm and the results achieved. Although the LKH algorithm, with its extremely high quality results, is more difficult to improve on than the CLK, nonetheless the multilevel framework was able to enhance the LKH performance. For example, in experiments on a well established test suite, the multilevel LKH scheme found 39 out of 59 optimal solutions as compared to the 33 found by LKH in a similar time period.