Improving polynomial bounds for the Graphical Traveling Salesman Problem with release dates on paths

The Graphical Traveling Salesman Problem with release dates (GTSP-rd) is a variation of the TSP-rd where each vertex in a weighted graph $G$ must be visited at least once, respecting the release date restriction. The edges may be traversed multiple times if necessary, as in some sparse graphs. This paper focuses on solving the GTSP-rd in paths. We consider two objective functions: minimizing the route completion time (GTSP-rd (time)) and minimizing the total distance traveled (GTSP-rd (distance)). We present improvements to existing dynamic programming algorithms, offering an $O(n)$ solution for paths where the depot is located at the extremity and an $O(n^2)$ solution for paths where the depot is located anywhere. For the GTSP-rd (distance), we propose an $O(n \log \log n)$ solution for the case with the depot at the extremity and an $O(n^2 \log \log n)$ solution for the general case.