An Algorithm to Solve the Euclidean Single-Depot Bounded Multiple Traveling Salesman Problem

In the Multiple Traveling Salesman Problem (MTSP) we are given
$n+1$
cities, among which there is one distinct city called depot and
$k$
salesmen. A tour for each salesman that starts from the depot, ends also in depot and visits once some other cities, is to be constructed. The cost of a tour is the sum of the individual costs of each pair of cities from that tour. There are lower and upper bounds on the number of cities that each salesman has to visit. Respecting these restrictions, we wish to minimize the total cost of all
$k$
tours. We study the two-dimensional Euclidean case and suggest a 3-phase heuristic algorithm. At phase 1 we partition the whole set of cities into
$k$
disjoint subsets. At phase 2 we built a feasible tour for each of these subsets. Phase 3 is designed to improve the tours of phase 2. Among the 22 known benchmark instances, our algorithm improved the best known solutions (BKS) for 12 of them, for 4 benchmark instances we obtained the earlier best known costs, for 4 of the instances our algorithm gave an approximation gap within 3% of the BKS, and for the remaining 2 instances the approximation gap was more than 3%.