  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.
