Abstract
In the path version of the Travelling Salesman Problem (Path-TSP), a salesman is looking for the shortest Hamiltonian path through a set of n cities. The salesman has to start his journey at a given city s, visit every city exactly once, and finally end his trip at another given city t.In this paper we show that a special case of the Path-TSP with a Demidenko distance matrix is solvable in polynomial time. Demidenko distance matrices fulfill a particular condition abstracted from the convex Euclidian special case by Demidenko (1979) as an extension of an earlier work of Kalmanson (1975). We identify a number of crucial combinatorial properties of the optimal solution and design a dynamic programming approach with time complexity O(n6).
Highlights
The travelling salesman problem (TSP) is one of the best studied problems in operational research
We show that we can optimize in polynomial time over the TSP-paths of that nicely structured form
Consider a Path-TSP with a Demidenko distance matrix and an optimal (1, t)-TSP-path τ which contains no forbidden pairs of arcs and such that its peaks decrease and its valleys increase from the left to the right in the path
Summary
The travelling salesman problem (TSP) is one of the best studied problems in operational research. In the Path-TSP the instance specifies two cities s and t (with s= t), and the goal is to find a shortest route starting at city s, ending at city t, and visiting all the other cities exactly once. Both the TSP and the Path-TSP are NP-hard to solve exactly (see for instance Garey & Johnson [12]), and both problems are APX-hard to approximate (Papadimitriou & Yannakakis [22]; Zenklusen [25]) These intractability results hold even in the metric case, where the distances between the cities are non-negative and satisfy the triangle inequality.
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