[Paper Review] Apply Local Clustering Method to Improve the Running Speed of Ant Colony Optimization

Chao-Yang Pang, Wei Hu+2 more|ArXiv.org|Jul 6, 2009

Metaheuristic Optimization Algorithms Research31 references3 citations

TL;DR

This paper proposes a novel Local Clustering method, Special Local Clustering (SLC), to drastically accelerate Ant Colony Optimization (ACO) for the Traveling Salesman Problem (TSP). By grouping cities into compact, tightly clustered regions, ACO operates on smaller subproblems, reducing both the number of cities (N) and iterations (t_max), achieving speedups of up to 10,736× while maintaining solution quality within 2% of standard ACO.

ABSTRACT

Ant Colony Optimization (ACO) has time complexity O(t*m*N*N), and its typical application is to solve Traveling Salesman Problem (TSP), where t, m, and N denotes the iteration number, number of ants, number of cities respectively. Cutting down running time is one of study focuses, and one way is to decrease parameter t and N, especially N. For this focus, the following method is presented in this paper. Firstly, design a novel clustering algorithm named Special Local Clustering algorithm (SLC), then apply it to classify all cities into compact classes, where compact class is the class that all cities in this class cluster tightly in a small region. Secondly, let ACO act on every class to get a local TSP route. Thirdly, all local TSP routes are jointed to form solution. Fourthly, the inaccuracy of solution caused by clustering is eliminated. Simulation shows that the presented method improves the running speed of ACO by 200 factors at least. And this high speed is benefit from two factors. One is that class has small size and parameter N is cut down. The route length at every iteration step is convergent when ACO acts on compact class. The other factor is that, using the convergence of route length as termination criterion of ACO and parameter t is cut down.

Motivation & Objective

• Reduce the high computational cost of Ant Colony Optimization (ACO), which scales as O(t_max * M * N²), especially due to large N.
• Address the key bottleneck in ACO: long running time caused by large problem size (N) and excessive iterations (t_max).
• Develop a clustering method that generates compact, tightly grouped city clusters to enable faster local optimization.
• Maintain solution quality by correcting inaccuracies introduced by clustering, particularly cross-edges and mismatched distributions.
• Propose and evaluate enhanced ACO variants—ACO-SLC-LWCR and ACO-SLC-Mixture—that balance speed and accuracy.

Proposed method

• Introduce a novel clustering algorithm called Special Local Clustering (SLC), which iteratively identifies and removes clusters with converging distortion, ensuring compactness.
• Use the convergence of route length (|L_t - L_{t+1}| / L_t → 0) as the termination criterion for ACO, replacing fixed t_max and reducing iterations.
• Apply ACO independently on each compact cluster to solve local TSPs, significantly reducing N per subproblem.
• Join local TSP routes into a global solution, with post-processing to eliminate cross-edges that degrade solution quality.
• Introduce ACO-SLC-LWCR to handle cross-edges via local reconnection, and ACO-SLC-Mixture to manage non-uniform city distributions.
• Leverage visual inspection for initial centroids in SLC to reduce sensitivity to initialization, improving clustering stability.

Experimental results

Research questions

• RQ1 Can clustering cities into compact, tightly grouped regions significantly reduce the running time of ACO for TSP?
• RQ2 Does using route length convergence as a termination criterion for ACO reduce the number of iterations (t_max) without sacrificing solution quality?
• RQ3 To what extent does the SLC clustering method improve ACO performance compared to standard ACO and ACO-K-Means?
• RQ4 How can solution inaccuracy caused by clustering—especially cross-edges and distribution mismatches—be effectively mitigated?
• RQ5 Can the proposed ACO-SLC variants achieve both high speed and high solution quality across diverse TSP instances?

Key findings

• The ACO-SLC algorithm achieves a speedup of 415 to 10,736 times over standard ACO, primarily due to reduced N and t_max.
• The ACO-SLC-LWCR variant achieves a speedup of 390 to 10,192 times, with improved handling of cross-edges.
• The ACO-SLC-Mixture variant achieves a speedup of 257 to 9,419 times and maintains solution quality, with inaccuracy ratio less than ACO in most cases and at most 2% worse.
• The convergence of route length (|L_t - L_{t+1}| / L_t → 0) serves as an effective, adaptive termination criterion, reducing t_max significantly.
• Clustering quality is sensitive to initial centroids, but visual selection of centroids in simulations mitigates this issue.
• The method is most effective when cities naturally form compact clusters, and the SLC algorithm outperforms K-Means in generating compact classes.