[Paper Review] Charting the Complexity Landscape of Waypoint Routing

Saeed Akhoondian Amiri, Klaus-Tycho Foerster+2 more|arXiv (Cornell University)|Apr 28, 2017

Advanced Graph Theory Research35 references3 citations

TL;DR

This paper provides a comprehensive complexity analysis of the waypoint routing problem in computer networks, where traffic must traverse a sequence of intermediate nodes (waypoints) subject to capacity constraints. It establishes NP-completeness for general cases while identifying polynomial-time solvable scenarios, particularly when waypoints are flow-conserving or the graph structure is restricted, offering both algorithmic insights and hardness boundaries for network service chaining and segment routing applications.

ABSTRACT

Modern computer networks support interesting new routing models in which traffic flows from a source s to a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network functions, to create innovative network services like service chains or segment routing. While the benefits and technological challenges of providing such routing models have been articulated and studied intensively over the last years, much less is known about the underlying algorithmic traffic routing problems. This paper shows that the waypoint routing problem features a deep combinatorial structure, and we establish interesting connections to several classic graph theoretical problems. We find that the difficulty of the waypoint routing problem depends on the specific setting, and chart a comprehensive landscape of the computational complexity. In particular, we derive several NP-hardness results, but we also demonstrate that exact polynomial-time algorithms exist for a wide range of practically relevant scenarios.

Motivation & Objective

• To analyze the computational complexity of routing a flow through a sequence of waypoints in modern networks, especially in the context of network function virtualization (NFV) and service chaining.
• To identify the conditions under which the waypoint routing problem is tractable or intractable, particularly concerning flow size changes and graph topology.
• To establish connections between waypoint routing and classic graph problems such as disjoint paths and cycle detection.
• To provide a foundational understanding of algorithmic challenges in modern network services that require traffic to pass through ordered network functions or middleboxes.

Proposed method

• Reduction from the Partition problem to prove NP-completeness for general waypoint routing with arbitrary flow size changes.
• Construction of graph instances using stars and cycles with carefully assigned capacities to simulate flow size changes at waypoints.
• Use of binary encoding of large flow sizes to demonstrate strong NP-completeness even when input size is polynomial in the logarithm of the flow values.
• Adaptation of the construction to directed graphs by replacing undirected edges with bidirectional edges of equal capacity.
• Introduction of dummy waypoints that increase flow size by a constant factor to preserve hardness under restricted flow-change models.
• Leveraging known results on disjoint paths and shortest cycles to contrast the complexity of unordered vs. ordered waypoint routing problems.

Experimental results

Research questions

• RQ1 Under what conditions is the waypoint routing problem with capacity constraints computationally tractable or intractable?
• RQ2 How does the ability of waypoints to alter flow size impact the complexity of finding a feasible or shortest routing path?
• RQ3 What is the computational complexity of waypoint routing in directed graphs, especially when flow size changes are bounded?
• RQ4 How do different graph topologies (e.g., undirected, directed, unicyclic) influence the complexity of the problem?
• RQ5 What is the relationship between waypoint routing and classical graph problems such as disjoint paths and cycle detection?

Key findings

• The waypoint routing problem is strongly NP-complete, even when flow sizes are encoded in binary and the underlying graph is unicyclic.
• The problem remains NP-complete for directed graphs when waypoints can only increase or decrease flow size by a constant factor.
• Polynomial-time algorithms exist for a wide range of practical scenarios, particularly when waypoints are flow-conserving or the number of waypoints is small.
• The complexity landscape depends critically on whether waypoints alter flow size and on the graph’s topological structure.
• There is a fundamental difference in complexity between ordered waypoint routing (this work) and unordered variants, with the former being significantly harder in general.
• The reduction from the Partition problem establishes that even with small, constant flow changes, the problem remains computationally hard.