A traveller's problem

A traveller is planning a tour from some start position, s, to a goal position g in d-dimensional space. Transportation is provided by n carriers. Each carrier is a convex object that results from intersecting finitely many closed linear subspaces; it moves at constant speed along a line. Different carriers may be assigned different velocity vectors. While using carrier C, the traveller can walk at innate speed v ≥ 0 in any direction, like a passenger on board a vessel. Whenever his current position on C is simultaneously contained in some other carrier C', the traveller can change from C to C', and continue his tour by C'. Given initial positions of the carriers and of s and g, is the traveller able to reach g starting from s? If so, what minimum travel time can be achieved? We provide the following answers. For a situation similar to the "Frogger" game, where the traveller has to cross a river on which n consecutive rectangular barges move at m different speeds, we provide an O(n log m) solution. In dimension 8 and higher, Traveller's Problem is undecidable, even for innate speed zero. An interesting case is in dimension 2. We prove that the problem is NP-hard, even if all carriers are vertical line segments. It turns out that an s-to-g path of finite duration may require an infinite number of carrier changes. Despite this difficulty, we can show that the two-dimensional problem is decidable. In addition, we provide a pseudo-polynomial approximation algorithm.