A General Solution to Bellman's Lost-in-a-forest Problem

TL;DR

This paper introduces a comprehensive framework for solving Bellman's lost-in-a-forest problem by transforming it into a constrained optimization problem, applicable to various related geometric problems.

Contribution

It provides a novel, general formulation and solution approach for the lost-in-a-forest problem and extends it to related geometric problems like Moser's worm problem.

Findings

Transforming the problem simplifies it into a constrained minimization.

Discretization links the problem to TSP and Hamiltonian path variants.

The approach is adaptable to multiple geometric problems.

Abstract

We present a general solution and formulation framework to Bellman's lost-in-a-forest problem. The forest boundary is known and may take any shape. The starting point and the orientation are unspecified. We convert the problem into translation and rotation of the forest boundary. This transformation allows us to formulate this problem as a constrained minimization problem. Upon discretization, the problem becomes a variation of the traveling salesman problem or the Hamiltonian path problem. We leverage discrete optimization and derive several nontrivial results consistent with those from previous papers. This method is general, and we also extend the approach to related problems, including Moser's worm problem and the shortest opaque set problem.