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

TL;DR

This paper improves algorithms for solving the Graphical Traveling Salesman Problem with release dates on paths, providing more efficient solutions for route time and distance minimization in specific path configurations.

Contribution

It introduces improved dynamic programming algorithms with better time complexities for GTSP-rd on paths, including cases with depot at extremity and arbitrary locations.

Findings

O(n) solution for route completion time with depot at extremity

O(n^2) solution for route completion time in general paths

O(n log log n) solution for minimizing total distance with depot at extremity

Abstract

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.