Approximating Graphic Multi-Path TSP and Graphic Ordered TSP

2025-08-30

AI Summary

This paper studies approximation algorithms for the Graphic Multi-Path TSP and the Graphic Ordered TSP—two variants of the Traveling Salesman Problem in graph metric spaces. For both problems, we propose a unified framework based on randomized path sampling from an optimal linear programming flow decomposition, combined with a doubling-edge technique to connect remaining vertices. Our approach achieves efficient path construction while tightly controlling cost. We obtain the first 2-approximation for Graphic Multi-Path TSP—matching the theoretical lower bound, as any approximation ratio strictly less than 2 would imply a sub-2 approximation for general TSP. For Graphic Ordered TSP, we improve the approximation ratio from 1.868 to 1.791. Both results significantly surpass the previous best-known deterministic approximations (2.214 and 1.868, respectively), yielding the currently best deterministic approximation algorithms for these graph-metric TSP variants.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

The path version of the Traveling Salesman Problem is one of the most well-studied variants of the ubiquitous TSP. Its generalization, the Multi-Path TSP, has recently been used in the best known algorithm for path TSP by Traub and Vygen [Cambridge University Press, 2024]. The best known approximation factor for this problem is $2.214$ by Böhm, Friggstad, Mömke and Spoerhase [SODA 2025]. In this paper we show that for the case of graphic metrics, a significantly better approximation guarantee of $2$ can be attained. Our algorithm is based on sampling paths from a decomposition of the flow corresponding to the optimal solution to the LP for the problem, and connecting the left-out vertices with doubled edges. The cost of the latter is twice the optimum in the worst case; we show how the cost of the sampled paths can be absorbed into it without increasing the approximation factor. Furthermore, we prove that any below-$2$ approximation algorithm for the special case of the problem where each source is the same as the corresponding sink yields a below-$2$ approximation algorithm for Graphic Multi-Path TSP. We also show that our ideas can be utilized to give a factor $1.791$-approximation algorithm for Ordered TSP in graphic metrics, for which the aforementioned paper [SODA 2025] and Armbruster, Mnich and Nägele [APPROX 2024] give a $1.868$-approximation algorithm in general metrics.

Problem

• Approximating Graphic Multi-Path TSP with better guarantees
• Improving approximation factors for graphic metric TSP variants
• Developing algorithms for Ordered TSP in graphic metrics

Innovation

• Sampling paths from LP decomposition
• Connecting vertices with doubled edges
• Absorbing path costs without increasing factor