[Paper Review] P = NP: Linear programming formulation of the traveling salesman problem

Moustapha Diaby|arXiv (Cornell University)|Sep 2, 2006

Vehicle Routing Optimization Methods10 references4 citations

TL;DR

This paper presents a polynomial-sized linear programming formulation for the Traveling Salesman Problem (TSP) using a network flow-based approach. The key contribution is a compact, exact linear program that captures TSP constraints efficiently, offering a potential pathway to solving TSP in polynomial time if the formulation is proven valid and solvable in practice.

ABSTRACT

In this paper, we present a polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). The proposed linear program is a network flow-based model. Numerical implementation issues and results are discussed. (The exposition and proofs are much more detailed in an edition which I wrote in collaboration with Dr. M.H. Karwan in 2012-2014 . That edition is available at this http URL)

Motivation & Objective

• To develop a polynomial-sized linear program that exactly models the Traveling Salesman Problem.
• To provide a network flow-based formulation that captures TSP constraints with reduced variable and constraint count.
• To demonstrate numerical feasibility and implementation considerations for the proposed formulation.
• To lay the groundwork for a potential proof that P = NP via a tractable linear programming formulation of TSP.

Proposed method

• The formulation uses a network flow structure to model TSP tours as feasible flows in a directed graph.
• It introduces binary variables for edges and flow variables to represent path traversal, ensuring connectivity and tour constraints.
• The model enforces subtour elimination using flow-based constraints rather than exponential families of inequalities.
• The formulation is designed to be polynomial in size relative to the number of cities, avoiding exponential constraint sets.
• Numerical implementation focuses on solving the linear program using standard solvers, with attention to conditioning and scalability.

Experimental results

Research questions

• RQ1 Can the TSP be formulated as a polynomial-sized linear program without relying on exponentially many constraints?
• RQ2 Does a network flow-based formulation capture all TSP tour constraints while remaining compact?
• RQ3 Is the proposed formulation numerically stable and solvable in practice with standard LP solvers?
• RQ4 Can this formulation serve as a basis for proving P = NP if it is both correct and solvable in polynomial time?

Key findings

• The proposed linear program is of polynomial size relative to the number of cities in the TSP instance.
• The formulation uses a network flow structure to implicitly enforce subtour elimination and connectivity.
• The model avoids the need for exponentially many subtour elimination constraints by embedding them in flow-based constraints.
• Numerical results suggest the formulation is implementable and solvable with standard linear programming solvers, though full scalability remains to be demonstrated.