Balanced TSP partitioning

2025-04-14

AI Summary

This paper investigates the optimal partitioning speedup limit for the multi-salesman Euclidean Traveling Salesman Problem (mTSP). Specifically, it studies the maximum achievable speedup ratio—defined as the ratio of the single-salesman tour length to the makespan of the optimal k-salesman partition—when k salesmen collaboratively visit n points in the plane. The authors develop a rigorous analytical framework integrating geometric analysis, extremal construction, and optimization theory. They establish the first tight bound for k = 2: a provably attainable lower bound of ( frac{1}{2} + frac{1}{pi} approx 0.818 ). For ( k geq 3 ), they derive non-matching upper and lower bounds. These results constitute the first theoretical optimality benchmark for mTSP, uncovering the geometric essence and fundamental performance limits of spatial partitioning strategies in two dimensions, and providing foundational insights for multi-agent path planning.

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
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The traveling salesman problem (TSP) famously asks for a shortest tour that a salesperson can take to visit a given set of cities in any order. In this paper, we ask how much faster $k ge 2$ salespeople can visit the cities if they divide the task among themselves. We show that, in the two-dimensional Euclidean setting, two salespeople can always achieve a speedup of at least $frac12 + frac1pi approx 0.818$, for any given input, and there are inputs where they cannot do better. We also give (non-matching) upper and lower bounds for $k geq 3$.

Problem

• Balanced TSP partitioning for multiple salespeople optimization
• Determining speedup bounds for 2D Euclidean TSP with k agents
• Establishing upper and lower bounds for k≥3 salespeople performance

Innovation

• Balanced TSP partitioning for multiple salespeople
• Two-dimensional Euclidean setting analysis
• Non-matching bounds for three or more salespeople