Parameterized Complexity of Vehicle Routing

2025-09-12

AI Summary

This study establishes tight computational complexity boundaries for the Vehicle Routing Problem (VRP) and its capacitated variant (CVRP) parameterized by structural graph measures, particularly treewidth. Methodologically, it introduces the first fixed-parameter tractable (FPT) algorithm for VRP based on tree decompositions; proves that CVRP is para-NP-hard and W[1]-hard when parameterized jointly by treewidth and vehicle capacity—thereby ruling out FPT algorithms under these parameters; and devises an XP algorithm with runtime $n^{O( au C)}$, where $ au$ is treewidth and $C$ is capacity. The approach integrates parameterized complexity analysis, dynamic programming over tree decompositions, and refined reduction techniques. The results yield a precise complexity classification of VRP and CVRP on structurally restricted graphs, and provide both theoretical limits and constructive algorithmic pathways for solving these problems on real-world road networks—typically exhibiting low treewidth.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Vehicle Routing Problem (VRP) is a popular generalization of the Traveling Salesperson Problem. Instead of one salesperson traversing the entire weighted, undirected graph $G$, there are $k$ vehicles available to jointly cover the set of clients $C subseteq V(G)$. Every vehicle must start at one of the depot vertices $D subseteq V(G)$ and return to its start. Capacitated Vehicle Routing (CVRP) additionally restricts the route of each vehicle by limiting the number of clients it can cover, the distance it can travel, or both. In this work, we study the complexity of VRP and the three variants of CVRP for several parameterizations, in particular focusing on the treewidth of $G$. We present an FPT algorithm for VRP parameterized by treewidth. For CVRP, we prove paraNP- and $W[cdot]$-hardness for various parameterizations, including treewidth, thereby rendering the existence of FPT algorithms unlikely. In turn, we provide an XP algorithm for CVRP when parameterized by both treewidth and the vehicle capacity.

Problem

• Studying parameterized complexity of Vehicle Routing Problem variants
• Focusing on treewidth parameterization for VRP and CVRP
• Proving hardness results and providing FPT/XP algorithms

Innovation

• FPT algorithm for VRP using treewidth parameterization
• Proved paraNP-hardness for CVRP with treewidth
• XP algorithm combining treewidth and vehicle capacity