In this paper, we consider the $k$-Covering Canadian Traveller Problem ($k$-CCTP), which can be seen as a variant of the Travelling Salesperson Problem. The goal of $k$-CCTP is finding the shortest tour for a traveller to visit a set of locations in a given graph and return to the origin. Crucially, unknown to the traveller, up to $k$ edges of the graph are blocked and the traveller only discovers blocked edges online at one of their respective endpoints. The currently best known upper bound for $k$-CCTP is $O(\sqrt{k})$ which was shown in [Huang and Liao, ISAAC '12]. We improve this polynomial bound to a logarithmic one by presenting a deterministic $O(\log k)$-competitive algorithm that runs in polynomial time. Further, we demonstrate the tightness of our analysis by giving a lower bound instance for our algorithm.

翻译：本文考虑 $k$-覆盖式加拿大旅行者问题（$k$-CCTP），该问题可视为旅行商问题的一种变体。$k$-CCTP 的目标是找出一条最短回路，使旅行者访问给定图中的一组位置并返回起点。关键之处在于，旅行者事先不知情，图中至多 $k$ 条边被阻塞，且旅行者仅能在到达其任一端点时在线发现这些阻塞边。当前 $k$-CCTP 已知的最佳上界为 $O(\sqrt{k})$，该结果由 [Huang and Liao, ISAAC '12] 给出。本文将该多项式上界改进为对数上界，提出一种运行于多项式时间内的确定性 $O(\log k)$-竞争算法。此外，我们通过给出该算法的下界实例，证明了分析的紧致性。