[Paper Review] A Novel Mathematical Model for the Unique Shortest Path Routing Problem

Ch. Zhang|arXiv (Cornell University)|Jun 30, 2008

Network Traffic and Congestion Control13 references3 citations

TL;DR

This paper proposes a novel origin-based mathematical model (OBM) for the unique shortest path routing problem, formulating it with a polynomial number of constraints to enable efficient exact solutions via decomposition methods. The OBM reduces model size and improves constraint structure compared to prior demand-based formulations, enabling faster convergence in Benders decomposition and outperforming existing approaches on large-scale network instances.

ABSTRACT

Link weights are the principal parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem properly, so that a solution algorithm may be developed which could prove to be more efficient than those already in existence. In this paper, a novel complete formulation with a polynomial number of constraints is first introduced and then mathematically proved to be correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem.

Motivation & Objective

• To develop a complete, exact mathematical formulation for the unique shortest path routing problem with polynomial-sized constraints.
• To address the limitations of prior heuristic and incomplete formulations that fail to guarantee optimality.
• To improve solution efficiency by enabling effective decomposition algorithms, particularly Benders decomposition.
• To compare the new formulation with an existing demand-based model in terms of constraint structure and model size.
• To provide a foundation for scalable, exact solutions to real-world network routing problems with average-sized data instances.

Proposed method

• Proposes an origin-based formulation (OBM) that restructures constraints around origin nodes rather than individual demands, reducing complexity.
• Mathematically proves the correctness and equivalence of OBM to the original problem in terms of feasibility and optimality.
• Introduces auxiliary flow variables and flow bound constraints to explicitly model path uniqueness and routing decisions.
• Demonstrates that OBM allows global decomposition into one master problem and two subproblems, enhancing algorithmic flexibility.
• Applies Benders decomposition to OBM, enabling independent subproblems per origin node and improving scalability.
• Compares OBM with a prior demand-based model (DBM) using model size and constraint structure metrics on a large-scale network instance.

Experimental results

Research questions

• RQ1 How can the unique shortest path routing problem be mathematically formulated with a polynomial number of constraints to ensure exact solvability?
• RQ2 What structural advantages does an origin-based formulation offer over a demand-based formulation in terms of decomposition efficiency?
• RQ3 How does the model size of the new formulation compare to existing formulations on large-scale network instances?
• RQ4 Can the new formulation support more efficient decomposition algorithms such as Benders decomposition?
• RQ5 What improvements in solution speed and scalability can be achieved through the new model structure?

Key findings

• The origin-based model (OBM) reduces the number of variables in the original problem from over 645,000 to 64,200 on a large instance with 50 nodes, 642 links, and 1,000 demands.
• The number of constraints in the original problem drops from over 1.3 million to 37,742 under OBM, representing a significant reduction.
• The master problem in OBM has 20 times fewer variables and constraints than the master problem in the demand-based model (DBM).
• OBM enables a more flexible decomposition structure—supporting one master and two subproblems—compared to DBM’s one master and one subproblem.
• The improved constraint structure and reduced model size of OBM enhance the efficiency of Benders decomposition and support faster convergence in practice.
• The paper establishes that OBM is both correct and equivalent to the original problem in terms of feasibility and optimality, validating its use as a robust solution framework.