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

2025-02-04

AI Summary

This paper investigates the Graph Traveling Salesman Problem with release dates (GTSP-rd) on path graphs, optimizing two objectives: minimizing the makespan (GTSP-rd(time)) and minimizing total travel distance (GTSP-rd(distance)). We propose the first dedicated algorithms for this setting: for the time objective, we design O(n) and O(n²) algorithms; for the distance objective, we present O(n log log n) and O(n² log log n) algorithms. Our approach integrates dynamic programming with divide-and-conquer optimization and acceleration via monotonic and priority queues, supporting arbitrary depot locations. Compared to standard DP approaches—whose time complexity is inherently Ω(n³) or higher—our algorithms break previous lower bounds, achieving theoretical optimality for path-structured instances. This work establishes the first near-linear scalable solution for real-time scheduling and routing optimization on paths, advancing both algorithmic theory and practical applicability in constrained graph routing.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Improving polynomial bounds for GTSP-rd
• Focusing on paths with release dates
• Enhancing dynamic programming algorithms

Innovation

• Improved dynamic programming algorithms
• Optimized route completion time
• Enhanced total distance efficiency