Algorithms for the On-Line Travelling Salesman1

Abstract

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On-Line Algorithms, Real Time, the Virtue of Laziness, and the Power of Clairvoyance

“…This bound shows that time lookahead is useless in the homing case, because a lower bound of 2 holds for the H-OL-Tsp even without lookahead, and an optimal on-line algorithm without lookahead exists [8]. The following proof extends an alternative proof of the 2-competitiveness of the H-OL-Tsp (without lookahead) given by Lipmann [35].…”

“…OL-Tsp has been introduced by Ausiello et al in [8]. In OL-Tsp we are given a metric space M = (X, d), where X is a set of points and d is a distance function on X, with a distinguished point O ∈ X, called the origin; and a set of requests σ = {σ 1 , .…”

“…In order to show some of the basic techniques used to deal with this kind of problems we now recall, from [8], the proofs of a lower bound and of an upper bound for the N-OL-Tsp in general metric spaces. Note that, as it can be seen from Table 1, these results have been improved in [35]; however we report them here because they are simple enough to provide a good introduction, while the ones in [35] Proof.…”

“…The following algorithm, which is derived from [8], is the best possible among zealous algorithms for the homing OL-A-Tsp.…”

“…Dealing with the H-OL-Tsp, Jaillet and Wagner [27] compare ∆ with the length of the optimal tour L T SP : they prove that, if ∆ = αL T SP , then there exists a 2 − α 1+α -competitive algorithm in the general metric space, that improves the 2-competitive algorithm of [8]. Notice that, as a consequence of Theorem 11, this result crucially depends on the fact that lookahead is not fixed a-priori, but depends on the input instance.…”