Two multi-start heuristics for the k -traveling salesman problem

This paper is concerned with the k-traveling salesman problem (k-TSP), which is a variation of widely studied traveling salesman problem (TSP). Given a set of n cities including a home city and a fixed value k such that $$1 < k \le n$$ , this problem seeks a tour of minimum length which starts and ends at the home city and visits k cities (including the home city) exactly out of these n cities. Finding a feasible solution to k-TSP involves finding a subset of k cities including the home city first, and then a circular permutation representing a tour of these k cities. In this paper, we have proposed two multi-start heuristic approaches for the k-TSP. The first approach is based on general variable neighborhood search algorithm (GVNS), whereas the latter approach is a hyper-heuristic (HH) approach. A variable neighborhood descent strategy operating over two neighborhood structures is utilized for doing the local search in the GVNS. As part of the hyper-heuristic, two low level heuristics are considered. To the best of our knowledge, these are the first metaheuristic and hyper-heuristic approaches for the k-TSP. To evaluate the performance of our approaches, a set of benchmark instances is created utilizing instances from TSPLIB. Computational results on these benchmark instances show HH approach to be better than GVNS approach.