On the Power of Lookahead in On-Line Vehicle Routing Problems

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

“…Allulli et al introduced request lookahead k and time lookahead ∆ [5,4]; independently, Jaillet and Wagner proposed the disclosures dates model [27], which is very close to time lookahead.…”

“…It is unrealistic to assume that a real-world application would dispose of request lookahead; furthermore, it is unlikely that the quite bizarre additional information provided by request lookahead yield meaningful performance improvement to on-line algorithms. Indeed, in [4] some vehicle routing problems are analyzed, concluding that the only advantage originated by request lookahead is the possibility of knowing, at any time, if the input instance is finished, i.e. if no more requests will be released in the future.…”

“…This kind of lookahead has natural applications: it is easy to conceive scenarios where, using a more sophisticated data collecting process, the on-line algorithm would learn requests with some fixed advance. In order to be meaningful, ∆ must be related to some characteristic quantities of the model: in [5] vehicle routing problems are considered only in limited metric spaces, and ∆ is compared to the time necessary for a server to traverse the entire metric space, i.e. the diameter D of the metric space (the server moves at unit speed).…”

“…We consider the model of Allulli et al: in [5] they prove a lower bound of 2 for both the Homing and the Nomadic OL-Tsp, in the general metric space. 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].…”

“…In [5] it is described an algorithm, ReturnHome α , whose competitive ratio improves with δ = ∆/D, matching the lower bound of 2 when δ ≥ 1 (i.e. when ∆ ≥ D).…”