Parameterized Complexity of Directed Traveling Salesman Problem

2025-06-27

AI Summary

This paper systematically investigates the parameterized complexity of the Directed Traveling Salesman Problem (DTSP) and its generalized variant, the Directed Waypoint Routing Problem (DWRP). Using parameterized algorithm design and hardness reductions, we analyze DWRP with respect to key structural parameters: solution size, feedback edge number, vertex integrity, treewidth, and distance to constant-tree-depth graphs. Our main contributions are: (i) DWRP is fixed-parameter tractable (FPT) parameterized by solution size, feedback edge number, or vertex integrity; (ii) it lies in XP when parameterized by treewidth; and (iii) it is W[1]-hard parameterized by distance to constant-tree-depth. This work establishes the first comprehensive parameterized complexity classification for DWRP, resolving a long-standing open problem in the parameterized analysis of DTSP-type problems and providing a foundational framework for understanding their computational boundaries.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Directed Traveling Salesman Problem (DTSP) is a variant of the classical Traveling Salesman Problem in which the edges in the graph are directed and a vertex and edge can be visited multiple times. The goal is to find a directed closed walk of minimum length (or total weight) that visits every vertex of the given graph at least once. In a yet more general version, Directed Waypoint Routing Problem (DWRP), some vertices are marked as terminals and we are only required to visit all terminals. Furthermore, each edge has its capacity bounding the number of times this edge can be used by a solution. While both problems (and many other variants of TSP) were extensively investigated, mostly from the approximation point of view, there are surprisingly few results concerning the parameterized complexity. Our starting point is the result of Marx et al. [APPROX/RANDOM 2016] who proved that DTSP is W[1]-hard parameterized by distance to pathwidth 3. In this paper we aim to initiate the systematic complexity study of variants of DTSP with respect to various, mostly structural, parameters. We show that DWRP is FPT parameterized by the solution size, the feedback edge number, and the vertex integrity of the underlying undirected graph. Furthermore, the problem is XP parameterized by treewidth. On the complexity side, we show that the problem is W[1]-hard parameterized by the distance to constant treedepth.

Problem

• Study parameterized complexity of Directed Traveling Salesman Problem (DTSP)
• Analyze Directed Waypoint Routing Problem (DWRP) with structural parameters
• Determine FPT and W[1]-hard cases for DWRP parameters

Innovation

• FPT parameterized by solution size
• FPT with feedback edge number
• XP parameterized by treewidth