Parameterized Complexity of Segment Routing

2025-01-07

AI Summary

This paper studies the capacitated segment routing multi-commodity flow problem: given a graph network and a set of communication demands, select at most $k$ intermediate landmarks per demand pair such that all resulting segment-routed paths respect edge capacity constraints. Focusing on realistic network topologies—particularly graphs of bounded treewidth—the work establishes, for the first time, that the problem remains NP-hard even when treewidth is constant and $k = 1$, thereby ruling out the existence of an FPT algorithm with runtime $f(d) cdot n^{g(k)}$. It further identifies practically relevant polynomial-time solvable special cases and provides the first rigorous theoretical lower bound and the first exact polynomial-time solvability condition. The analysis integrates parameterized complexity theory, structural graph properties (especially treewidth), NP-hardness reductions, and fixed-parameter tractability criteria. Collectively, these results systematically uncover the fundamental computational bottlenecks of segment routing and deliver critical theoretical boundaries and promising avenues for designing efficient routing algorithms tailored to real-world networks.

Technology Category

• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

Segment Routing is a recent network technology that helps optimizing network throughput by providing finer control over the routing paths. Instead of routing directly from a source to a target, packets are routed via intermediate waypoints. Between consecutive waypoints, the packets are routed according to traditional shortest path routing protocols. Bottlenecks in the network can be avoided by such rerouting, preventing overloading parts of the network. The associated NP-hard computational problem is Segment Routing: Given a network on $n$ vertices, $d$ traffic demands (vertex pairs), and a (small) number $k$, the task is to find for each demand pair at most $k$ waypoints such that with shortest path routing along these waypoints, all demands are fulfilled without exceeding the capacities of the network. We investigate if special structures of real-world communication networks could be exploited algorithmically. Our results comprise NP-hardness on graphs with constant treewidth even if only one waypoint per demand is allowed. We further exclude (under standard complexity assumptions) algorithms with running time $f(d) n^{g(k)}$ for any functions $f$ and $g$. We complement these lower bounds with polynomial-time solvable special cases.

Problem

• Complex Network Structures
• Segment Routing
• Demand Satisfaction Complexity

Innovation

• Segment Routing
• Complexity Analysis
• Algorithmic Impossibility