Search and Delivery Man Problems: When Are Depth-First Paths Optimal?

TL;DR

This paper investigates conditions under which depth-first search paths are optimal in search and delivery problems on networks, providing new insights and correcting previous errors in the literature.

Contribution

It establishes sufficient and necessary conditions for depth-first paths to be optimal and corrects an error regarding depot location in the Delivery Man Problem.

Findings

Identifies when depth-first paths are optimal in tree networks.

Provides conditions on probability measures for optimality.

Corrects previous misconceptions about depot placement in delivery problems.

Abstract

Let h be a probability measure on the nodes and arcs of 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. We concentrate mainly on the search problem, with the Delivery Man Problem arising as a special case.