# travelling salesman problem

(redirected from Euclidean TSP)

## travelling salesman problem

(algorithm, complexity)
(TSP or "shortest path", US: "traveling") Given a set of towns and the distances between them, determine the shortest path starting from a given town, passing through all the other towns and returning to the first town.

This is a famous problem with a variety of solutions of varying complexity and efficiency. The simplest solution (the brute force approach) generates all possible routes and takes the shortest. This becomes impractical as the number of towns, N, increases since the number of possible routes is !(N-1). A more intelligent algorithm (similar to iterative deepening) considers the shortest path to each town which can be reached in one hop, then two hops, and so on until all towns have been visited. At each stage the algorithm maintains a "frontier" of reachable towns along with the shortest route to each. It then expands this frontier by one hop each time.

Pablo Moscato's TSP bibliography. Fractals and the TSP.
References in periodicals archive ?
Though the general Euclidean TSP is still NP-hard (Sanjeev [8]), there are some cases that are well solvable.
Similar to Euclidean TSP, although the general TSP on permuted Monge matrices is not polynomial solvable, a number of specific forms of matrices provide the possibility of polynomial solving algorithm.
Sanjeev, "Polynomial time approximation schemes for Euclidean TSP and other geometric problems," in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pp.
We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions.
Unfortunately, even Euclidean TSP is NP-hard (Papadimitriou [1977]; Garey et al.
The status of Euclidean TSP remained open, however.
Lemma 3 is implicit in prior work on Euclidean TSP [Beardwood et al.
THEOREM 5 (STRUCTURE THEOREM FOR EUCLIDEAN TSP IN [R.
The dynamic programming algorithm for Euclidean TSP in [R.
Since Euclidean TSP is strongly NP-hard (this follows from the reductions in Papadimitriou [1977], for example), any algorithm that computes (1 + c)-approximations must have a running time with some exponential dependence on c (unless, of course, NP-complete problems can be solved in subexponential time).

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