[Paper Review] State transition algorithm for traveling salesman problem

Chunhua Yang, Xiaolin Tang+2 more|arXiv (Cornell University)|Jun 1, 2012

Metaheuristic Optimization Algorithms Research3 references3 citations

TL;DR

This paper proposes a discrete state transition algorithm (STA) for the traveling salesman problem (TSP) using three specialized operators—swap, shift, and symmetry transformations—to efficiently explore the permutation-based solution space. The method achieves superior performance with faster convergence and lower computational time than simulated annealing and ant colony optimization, demonstrating strong search capability and stability without requiring parameter tuning.

ABSTRACT

Discrete version of state transition algorithm is proposed in order to solve the traveling salesman problem. Three special operators for discrete optimization problem named swap, shift and symmetry transformations are presented. Convergence analysis and time complexity of the algorithm are also considered. To make the algorithm simple and efficient, no parameter adjusting is suggested in current version. Experiments are carried out to test the performance of the strategy, and comparisons with simulated annealing and ant colony optimization have demonstrated the effectiveness of the proposed algorithm. The results also show that the discrete state transition algorithm consumes much less time and has better search ability than its counterparts, which indicates that state transition algorithm is with strong adaptability.

Motivation & Objective

• To address the challenge of solving the NP-hard TSP using a novel heuristic approach tailored for discrete permutation spaces.
• To design specialized state transition operators that efficiently navigate the combinatorial solution space of TSP.
• To develop a parameter-free strategy for the algorithm to enhance simplicity and robustness.
• To evaluate the algorithm’s performance against established metaheuristics like simulated annealing and ant colony optimization.

Proposed method

• The discrete state transition algorithm uses a unified state transition form where the next state is generated via a transformation matrix applied to the current solution sequence.
• Three specialized operators—swap, shift, and symmetry transformations—are introduced to manipulate permutations while preserving solution validity.
• The swap operator exchanges elements at randomly selected positions, the shift operator cyclically shifts a subsequence, and the symmetry operator reverses a subsequence.
• The algorithm employs a greedy selection criterion to accept only improving or equal-quality solutions, avoiding the need for temperature or pheromone parameter tuning.
• The transformation matrices are derived from general elementary matrices, ensuring one-to-one mapping between input and output permutations.
• The method is implemented without any adjustable parameters, simplifying deployment and enhancing stability.

Experimental results

Research questions

• RQ1 Can a discrete version of the state transition algorithm effectively solve the traveling salesman problem using specialized permutation operators?
• RQ2 How does the performance of the proposed STA compare to simulated annealing and ant colony optimization in terms of solution quality and convergence speed?
• RQ3 Does the absence of parameter tuning in the STA affect its robustness and search capability on TSP instances?
• RQ4 To what extent do the designed operators (swap, shift, symmetry) contribute to escaping local optima and avoiding premature convergence?
• RQ5 How does the algorithm scale in terms of computational time and solution quality across different TSP benchmark instances?

Key findings

• The discrete STA achieved the best-known solution for the att48 instance with a route length of 33,724, outperforming both simulated annealing and ant colony optimization.
• For the berlin52 instance, STA found the optimal solution with a route length of 7,544.4, and consumed only 1/40 and 1/35 of the time required by SA and ACO, respectively.
• The standard deviation of STA results was consistently near zero, indicating high stability across multiple runs.
• STA showed faster convergence than SA and ACO, with a steadily decreasing fitness curve that avoided stagnation and premature convergence.
• In all tested instances (ulysses16, att48, berlin52), the worst solution found by STA was better than the best solution found by ACO, highlighting its superior search capability.
• The algorithm consumed significantly less time than both SA and ACO—approximately half the time of SA and 1/9 to 1/30 of ACO’s time—demonstrating strong computational efficiency.