The Canadian Tour Operator Problem on paths: tight bounds and resource augmentation

In the prize-collecting travelling salesman problem, we are given a weighted graph G=(V,E) with edge weights ℓ :E→ℝ_+ , a special vertex r∈ V , penalties π :V→ℝ_+ and the goal is to find a closed tour T such that r∈ V(T) and such that the cost ℓ (T)+π (V∖ V(T)) , which is the sum of the edges in the tour and the cost of the vertices not spanned by T , is minimized. We consider an online variant of the prize-collecting travelling salesman problem related to graph exploration. In the Canadian Tour Operator Problem the task is to find a closed route for a tourist bus in a given network G=(V,E) in which some edges are blocked by avalanches. An online algorithm learns from a blocked edge only when reaching one of its endpoints. The bus operator has the option to avoid visiting each node v∈ V by paying a refund of π (v) to the tourists. The goal consists of minimizing the sum of the travel costs and the refunds. We study the problem on a simple (weighted) path and prove tight bounds on the competitiveness of deterministic algorithms. Specifically, we give an algorithm with competitive ratio equal to the golden ratio ϕ =(1+√(5))/2 . We also study the effect of resource augmentation, where the online algorithm either pays a discounted cost for traversing edges or for the penalties.