The Traveling Salesman Problem (TSP) aims at finding the shortest trip for a salesman, who has to visit each of the locations from a given set exactly once, starting and ending at the same location. Here, we consider the Euclidean version of the problem, in which the locations are points in the two-dimensional Euclidean space and the distances are correspondingly Euclidean distances. We propose simple, fast, and easily implementable heuristics that work well, in practice, for large real-life problem instances. The algorithm works on three phases, the constructive, the insertion, and the improvement phases. The first two phases run in time O(n 2 ) and the number of repetitions in the improvement phase, in practice, is bounded by a small constant. We have tested the practical behavior of our heuristics on the available benchmark problem instances. The approximation provided by our algorithm for the tested benchmark problem instances did not beat best known results. At the same time, comparing the CPU time used by our algorithm with that of the earlier known ones, in about 92% of the cases our algorithm has required less computational time. Our algorithm is also memory efficient: for the largest tested problem instance with 744,710 cities, it has used about 50 MiB, whereas the average memory usage for the remained 217 instances was 1.6 MiB.
The Traveling Salesman Problem (TSP) aims to find the shortest tour for a salesman who starts and ends in the same city and visits the remaining n−1 cities exactly once. There are a number of common generalizations of the problem including the Multiple Traveling Salesman Problem (MTSP), where instead of one salesman, there are k salesmen and the same amount of individual tours are to be constructed. We consider the Euclidean version of the problem where the distances between the cities are calculated in two-dimensional Euclidean space. Both general the TSP and its Euclidean version are strongly NP-hard. Hence, approximation algorithms with a good practical behavior are of primary interest. We describe a general method for the solution of the Euclidean versions of the TSP (including MTSP) that yields approximation algorithms with a favorable practical behavior for large real-life instances. Our method creates special types of convex hulls, which serve as a basis for the constructions of our initial and intermediate partial solutions. Here, we overview three algorithms; one of them is for the bounded version of the MTSP. The proposed novel algorithm for the Euclidean TSP provides close-to-optimal solutions for some real-life instances.
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