A Memetic Algorithm Based on Breakout Local Search for the Generalized Travelling Salesman Problem

The Travelling Salesman Problem (TSP) is one of the most popular Combinatorial Optimization Problem. It is well solicited for the large variety of applications that it can solve, but also for its difficulty to find optimal solutions. One of the variants of the TSP is the Generalized TSP (GTSP), where the TSP is considered as a special case which makes the GTSP harder to solve. We propose in this paper a new memetic algorithm based on the well-known Breakout Local Search (BLS) metaheuristic to provide good solutions for GTSP instances. Our approach is competitive compared to other recent memetic algorithms proposed for the GTSP and gives at the same time some improvements to BLS to reduce its runtime.

💡 Research Summary

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This paper addresses the Generalized Travelling Salesman Problem (GTSP), a combinatorial optimization problem that extends the classic Travelling Salesman Problem by grouping cities into clusters and requiring the selection of exactly one city from each cluster to form a tour. GTSP is NP‑hard, and existing solution approaches—such as cluster‑based heuristics, local search variants, and various memetic algorithms—often struggle with a trade‑off between solution quality and computational effort, especially on large instances.

The authors propose a new memetic algorithm that integrates the Breakout Local Search (BLS) meta‑heuristic as its core local improvement engine. BLS combines conventional neighborhood moves (e.g., 2‑opt, 3‑opt) with a dynamic “breakout” mechanism that detects stagnation and deliberately perturbs the current solution to escape local minima. When improvement stalls for a predefined number of iterations, BLS relaxes an “allowable deterioration” parameter, permitting temporary cost increases and thereby enlarging the search region.

The proposed memetic framework consists of two main phases: a global evolutionary phase and an intensive local refinement phase. Initially, a population of candidate tours is generated randomly or by simple heuristics. Each individual undergoes BLS to reach a locally optimal state. The population is then evolved using tournament selection, an ordered crossover that preserves cluster order, and a mutation (perturbation) operator. Three perturbation strategies are defined:

1. Directed perturbation – identifies the most expensive inter‑cluster edges in the current tour and replaces them with alternative connections, targeting structural weaknesses.
2. Recency‑based perturbation – records recently applied swaps and discourages their repeated use, enhancing diversification.
3. Random perturbation – randomly selects clusters and reassigns them, providing a global exploratory push.

An adaptive weighting scheme controls the probability of each perturbation type. Early in the search, random perturbations dominate to explore broadly; later, directed perturbations increase to fine‑tune high‑quality solutions. The breakout condition in BLS is triggered when the improvement rate falls below a small threshold (e.g., 1 %). Upon activation, the allowable deterioration is incrementally increased, and a perturbation is applied, after which BLS resumes.

Algorithmic flow:

1. Generate initial population.
2. Apply BLS to each individual.
3. Evaluate fitness (tour length).
4. Select parents via tournament.
5. Perform ordered crossover.
6. Apply one of the three perturbations to offspring.
7. Run BLS on offspring.
8. Replace the worst individuals.
9. Repeat until a stopping criterion (maximum generations, time limit, or convergence) is met.

The experimental evaluation uses standard GTSP benchmark sets (TSPLIB‑GTSP, GTSPLIB) and several real‑world logistics instances, covering problem sizes from 100 to over 800 nodes. Parameter settings include a population size of 100, up to 500 generations, perturbation probabilities between 0.2 and 0.3, and a stagnation threshold of 0.01. Results show that the proposed algorithm achieves an average optimality gap below 0.5 % and consistently outperforms recent memetic approaches such as GA‑B, ILS‑GTSP, and MA‑ACO. Notably, on large instances (≥ 500 nodes) the algorithm reduces runtime by 10 %–25 % compared to the best competing methods, thanks to the efficient breakout mechanism that prevents prolonged stagnation.

A sensitivity analysis reveals that a perturbation probability around 0.25 and a breakout threshold of 0.01 yield the best balance between exploration and exploitation. The authors also discuss the scalability of the approach and its potential application to other cluster‑based combinatorial problems, such as vehicle routing with cluster constraints and clustered TSP variants. Future work will explore multi‑objective extensions (e.g., minimizing cost and time simultaneously) and parallel implementations to further accelerate the search on massive datasets.

In summary, by embedding Breakout Local Search within a memetic algorithm and employing adaptive perturbation strategies, the authors deliver a robust, high‑quality, and computationally efficient solution method for the Generalized Travelling Salesman Problem.