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

Abstract

In the prize-collecting travelling salesman problem, we are given a weighted graph $G=(V,E)$ with edge weights $\ell :E\rightarrow \mathbb {R}_+$, a special vertex $r\in V$, penalties $\pi :V\rightarrow \mathbb {R}_+$ and the goal is to find a closed tour $T$ such that $r\in V(T)$ and such that the cost $\ell (T)+\pi (V\setminus V(T))$, 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 $v\in V$ by paying a refund of $\pi (v)$ 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 $\phi =(1+\sqrt{5})/2$. We also study the effect of resource augmentation, where the online algorithm either pays a discounted cost for traversing edges or for the penalties.