Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem

Abstract An O( n 3 ) heuristic algorithm is described for solving d -city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition. The algorithm involves as substeps the computation of a shortest spanning tree of the graph G defining the TSP and the finding of a minimum cost perfect matching of a certain induced subgraph of G . A worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3/2. This represents a 50% reduction over the value 2 which was the previously best known such ratio for the performance of other polynomial growth algorithms for the TSP.