Search and Delivery Man Problems: When are depth-first paths optimal?

Abstract

Let h be a probability measure on a network Q, viewed either as the location of a hidden object to be found or as the continuous distribution of customers receiving packages. We wish to find a trajectory starting from a specified root, or depot O that minimizes the expected search or delivery time. We call such a trajectory optimal. When Q is a tree, we ask for which h there is an optimal trajectory that is depth-first, and we find sufficient conditions and in some cases necessary and sufficient conditions on h. A consequence of our analysis is a determination of the optimal depot location in the Delivery Man Problem, correcting an error in the literature.A person, or more generally a target, is lost in a network. We know the lengths of the arcs of the network, and we have a belief about the probability distribution (or hiding distribution) according to which the target is hidden. We wish to choose a unit speed trajectory (or search) starting from a given point (the root) with the aim of minimizing the expected search time:that is the expected time to find the target. The target might be a child lost in a cave, as in the recent rescue in Thailand; or it might be a leak in a pipeline. The Searcher could be an underwater diver or an unmanned aerial vehicle (UAV), respectively. We call a search that minimizes the expected search time optimal. The target does not have to be at a node; it can be anywhere in the interior of any arc. This paper considers the problem when the network is a tree.

Search for an immobile hider on a stochastic network

“…This article should also be put into perspective with Alpern and Lidbetter (2019b) which deals with the question of knowing when depth-first search is optimal. Our article brings some elements to answer this question in the stochastic setting.…”