A Clustering-Enhanced Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem
Abstract
:1. Introduction
1.1. Literature Review
- Exact algorithms: These algorithms can theoretically find a global optimal solution. For instance, Assad and Xu [7] proposed an accurate branch-and-bound algorithm and gave the optimal solution for cases with a maximum number of vertices of 15. Cordone et al. [8] improve the Lagrangian branch-and-bound algorithm in [7]. Pereira et al. [9] introduced two parallel branch-and-bound algorithms embedded with two lower bounding procedures based on Lagrangian relaxation and solved instances, including some with 50 vertices. Rostami and Malucelli [10] developed new lower bounds for instances with up to 50 vertices based on a reformulation scheme and some new mixed 0–1 linear formulations. Exact algorithms are also presented for related QMSTP variants. Guimarães et al. [11] investigated semidefinite programming lower bounds and introduced a new branch-and-bound algorithm, which stands out as the best exact algorithm. Sotirov et al. [12] reported a hierarchy of lower bounds and derived a sequence of improved relaxations with increasing complexity. However, since QMSTP is an NP-hard problem [7], the search space grows exponentially with the number of vertices. With the expansion of the graph, an accurate branch and bound algorithm [7] and its improved algorithms [8,9,10,11,12] deteriorate rapidly.
- Population-based algorithms: These algorithms perform searches with multiple initial points in a parallel style and associated search operators [13]. Zhou et al. [14] proposed a genetic algorithm using Prüfer number coding, which increased the number of vertices that can be solved to 50 for the first time. However, Prüfer number coding can only be applied to complete graphs and lacks local search capability as well as heritability. For this reason, Palubeckis et al. [3] used edge-set coding to implement a hybrid genetic algorithm with neighborhood search. Sundar et al. [15] proposed an artificial bee colony algorithm (ABC), which assigned employed bees to exploit the neighborhood of a food resource, onlooker bees to explore nearby food resources and scout bees to introduce new food resources, and finally applied local search to further optimize the solution.
- Local search algorithms: These algorithms starts with a single initial solution and use iterative local search to obtain better solutions. Öncan et al. [16] proposed a local search algorithm that alternates between local search and random moves. Palubeckis et al. [3] proposed an iterative tabu search (ITS) algorithm and achieved the best results in comparison with multistart simulated annealing algorithm and hybrid genetic algorithm on instances introduced by Cordone and Passeri [17]. Cordone et al. [8] applied the improved data structure and neighborhood search to the tabu search algorithm and started a new round of search with a randomly generated new solution after several iterations. The algorithm HSII proposed by Lozano et al. [18] combines iterative tabu search and oscillation strategy, and its neighborhood search is characterized by preferentially replacing edges that contribute the most to the current solution cost. Fu et al. [19] proposed a three-phase search approach (TPS), which includes a descent neighborhood search phase to reach a local optimum, a local optima exploring phase to discover nearby local optima within a given regional area and a perturbation-based diversification phase to jump out of the current regional search area. The algorithm achieved the best results in the literature.
1.2. Contribution
First, we introduce a clustering mechanism to the edges of the graph, where edges can be grouped with short “distance” (total cost) in the same cluster. This mechanism can effectively guide the population initialization. Second, we adopt a tabu-based nearby exploration phase to explore the restricted surrounding areas to locate other local optima, which can guarantee the search intensification around chosen high-quality local optima. Third, we design a new combination operator and a new mutation operator. By inheriting valuable edges, the combination operator can generate promising and diversified offspring solutions that are used as starting points for local refinement. The mutation operator, which is designed based on Lévy distribution, can explore unvisited search regions, and it can prevent the population from premature. Finally, we integrate the above three parts as an overall solution, where all parts coordinate and promote each other to find the optima. Additionally, CMA reports improved upper bounds for the 25 most challenging benchmark instances with unknown optimal solutions, which are valuable references for future research on this problem.
2. Problem Description
- objective function:
- subject to:
3. Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem
3.1. Main Scheme
Algorithm 1: General architecture of CMA for the QMSTP |
3.2. Solution Representation
3.3. Population Initialization with Edge Clustering Mechanism
3.3.1. Edge Clustering Mechanism
Algorithm 2: Edge clustering |
3.3.2. Initial Solution Generation
Algorithm 3: Initial population generation from edge clusters |
3.4. Local Refinement
Algorithm 4: Local refinement |
3.4.1. Move Operator
3.4.2. Evaluation Technique
3.5. Tabu-Based Nearby Exploration
Algorithm 5: Tabu-based nearby exploration |
3.5.1. Tabu Search
3.5.2. Tabu List Management
3.6. Three-Parent Combination Operator
3.7. Mutation Operator
3.8. Population Management
3.9. Computational Complexity of the Proposed Algorithm
4. Computational Experiments
4.1. Benchmark
4.2. Experimental Settings
4.3. Edge Clustering Experiments
4.4. Experimental Results on Benchmark SS
4.5. Experimental Results on Benchmark RAND and SOAK
5. Analyses and Discussions
5.1. Impact of Edge Clustering on Convergence Speed
5.2. Impact of Three-Parent Combination Operator
5.3. Parameter Sensitivity Analysis
5.4. Discussions
6. Conclusions
6.1. Limitations
6.2. Future Research Perspectives
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameter | Section | Description | Default Value |
---|---|---|---|
r | 3.3.1 | cluster ratio | 1.2 |
3.5 | max allowed iterations without improvement | 4 | |
3.5.1 | exploration length | ||
3.5.2 | tabu tenures | ||
3.6 | probability of parent selecting | ||
3.7 | mutation parameter |
25 | 0.002 | 7 | 63.4 |
50 | 0.03 | 15 | 57.7 |
100 | 0.9 | 29 | 52.7 |
150 | 7 | 46 | 48.2 |
200 | 30 | 61 | 44.8 |
250 | 98 | 66 | 40.5 |
Index | Results | ABC [15] | QMTS-TS [8] | TPS [19] | TPS * | CMA | |
---|---|---|---|---|---|---|---|
25 | 1 | 5085 | 5085 | 5085 | 5085 | 5085 | |
5085.85 | - | - | 5085 | 5085 | |||
3.71 | - | - | 0 | 0 | |||
0.91 | 0.33 | 0.02 | 0.02 | 0.04 | |||
2 | 5081 | 5081 | 5081 | 5081 | 5081 | ||
5101.2 | - | - | 5081 | 5081 | |||
6.64 | - | - | 0 | 0 | |||
1.02 | 0.33 | 0.03 | 0.04 | 0.04 | |||
3 | 4962 | 4962 | 4962 | 4962 | 4962 | ||
4962 | - | - | 4962 | 4962 | |||
0 | - | - | 0 | 0 | |||
1.05 | 0.35 | 0.02 | 0.02 | 0.04 | |||
50 | 1 | 21,126 | 21,126 | 21,126 | 21,126 | 21,126 | |
21,157.25 | - | - | 21,126 | 21,126 | |||
34.4 | - | - | 0 | 0 | |||
8.68 | 2.52 | 0.28 | 0.25 | 0.3 | |||
2 | 21,123 | 21,106 | 21,106 | 21,106 | 21,106 | ||
21,179.85 | - | - | 21,106 | 21,106 | |||
32.47 | - | - | 0 | 0 | |||
8.84 | 2.52 | 0.28 | 0.41 | 0.3 | |||
3 | 21,059 | 21,059 | 21,059 | 21,059 | 21,059 | ||
21,091.95 | - | - | 21,059 | 21,059 | |||
59.67 | - | - | 0 | 0 | |||
9.51 | 2.53 | 0.21 | 0.23 | 0.3 | |||
100 | 1 | 89,098 | 88,871 | 88,745 | 88,944 | 88,701 | |
89,404.6 | - | - | 89,086.56 | 88,760.89 | |||
167.87 | - | - | 41.65 | 20.8 | |||
116.66 | 48.29 | 4.06 | 8.65 | 6.55 | |||
2 | 89,202 | 89,049 | 88,911 | 88,894 | 88,843 | ||
89,520.45 | - | - | 88,991.78 | 88,866.11 | |||
190.05 | - | - | 63.62 | 26.58 | |||
115.95 | 47.88 | 3.82 | 7.42 | 6.72 | |||
3 | 89,007 | 88,720 | 88,659 | 88,658 | 88,627 | ||
89,242.6 | - | - | 88,872.89 | 88,735.33 | |||
138.59 | - | - | 196.38 | 73.52 | |||
98.88 | 48.06 | 4.38 | 8.78 | 6.14 | |||
150 | 1 | 205,619 | 205,615 | 204,995 | 205,083 | 204,937 * | |
206,404.3 | - | - | 205,426.11 | 205,168 | |||
405.97 | - | - | 197.99 | 90.87 | |||
444.87 | 146.44 | 25.53 | 40.77 | 34.79 | |||
2 | 205,874 | 205,509 | 205,219 | 205,185 | 205,034 * | ||
206,300.55 | - | - | 205,404.22 | 205,183.56 | |||
243.36 | - | - | 142.27 | 55.62 | |||
374.33 | 146.15 | 24.69 | 38.11 | 35.86 | |||
3 | 205,634 | 205,094 | 205,076 | 205,055 | 205,028 * | ||
206,160.1 | - | - | 205,406.67 | 205,196.78 | |||
316.07 | - | - | 265.88 | 125.89 | |||
432.93 | 146.43 | 29.84 | 31.35 | 32.77 | |||
200 | 1 | 371,797 | 371,492 | 370,873 | 370,818 | 370,715 * | |
372,527.6 | - | - | 371,215.22 | 371,008.33 | |||
381.44 | - | - | 312.33 | 183.43 | |||
1141.42 | 316.02 | 75.98 | 69.28 | 57.89 | |||
2 | 371,864 | 371,698 | 370,853 | 370,825 | 370,824 * | ||
372,306.6 | - | - | 371,472.11 | 371,170.33 | |||
311.74 | - | - | 395.34 | 194.92 | |||
1155.6 | 316.61 | 69.32 | 59.83 | 55.44 | |||
3 | 372,156 | 371,584 | 370,954 | 370,943 | 370,901 * | ||
372,842.9 | - | - | 371,206 | 371,085.67 | |||
735.21 | - | - | 191.9 | 146.81 | |||
1276.71 | 316.22 | 64.14 | 64.26 | 56.21 | |||
250 | 1 | 587,924 | 586,861 | 586,265 | 586,196 | 586,171 * | |
588,785.1 | - | - | 586,809 | 586,718.67 | |||
578.65 | - | - | 489.5 | 315.59 | |||
2563.41 | 478.62 | 145.83 | 151.16 | 124.8 | |||
2 | 588,068 | 587,607 | 586,778 | 586,757 | 586,514 * | ||
588,731.45 | - | - | 586,907.44 | 586,782.89 | |||
368.08 | - | - | 272.22 | 181.11 | |||
2840.91 | 479.64 | 113.17 | 140.53 | 126.7 | |||
3 | 587,883 | 587,281 | 585,851 | 585,851 | 585,783 * | ||
588,534.95 | - | - | 586,683.89 | 586,415.44 | |||
463.2 | - | - | 550.31 | 329.49 | |||
2328.29 | 480.06 | 135.5 | 157.12 | 118.9 |
Index | Results | ITS [3] | HSII [18] | TPS [19] | TPS * | CMA | |
---|---|---|---|---|---|---|---|
150 | 1 | 192,946 | 192,606 | 192,369 | 192,427 | 192,329 * | |
193,244.5 | 192,910.1 | - | 192,688.8 | 192,576.3 | |||
- | - | - | 157.08 | 126.07 | |||
400 | 400 | 400 | 400 | 400 | |||
2 | 193,034 | 192,607 | 192,579 | 192,558 | 192,460 * | ||
193,369.9 | 192,922.8 | - | 192,702.8 | 192,628.1 | |||
- | - | - | 135.65 | 129.37 | |||
400 | 400 | 400 | 400 | 400 | |||
3 | 192,965 | 192,577 | 192,046 | 192,269 | 192,008 * | ||
193,303.1 | 192,792.6 | - | 192,657.8 | 192,490.1 | |||
- | - | - | 193.34 | 163.36 | |||
400 | 400 | 400 | 400 | 400 | |||
200 | 1 | 351,216 | 350,517 | 350,321 | 350,394 | 350,297 * | |
351,787.2 | 351,023.6 | - | 350,878.8 | 350,724 | |||
- | - | - | 201.72 | 158.01 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
2 | 351,312 | 350,389 | 350,231 | 350,576 | 350,446 | ||
351,823.7 | 350,902.4 | - | 350,918.4 | 350,750.6 | |||
- | - | - | 236.06 | 163.72 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
3 | 351,466 | 351,057 | 350,601 | 350,677 | 350,544 * | ||
351,940.8 | 351,285.4 | - | 350,929.2 | 350,809.6 | |||
- | - | - | 243.37 | 182.45 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
250 | 1 | 558,451 | 556,929 | 556,596 | 556,897 | 556,588 * | |
559,235.5 | 557,434.6 | - | 557,689.4 | 557,363.2 | |||
- | - | - | 352.75 | 326.27 | |||
2000 | 2000 | 2000 | 2000 | 2000 | |||
2 | 558,820 | 557,474 | 556,604 | 556,823 | 556,598 * | ||
559,478.2 | 557,850.1 | - | 557,571.8 | 557,356.6 | |||
- | - | - | 389 | 315.1 | |||
2000 | 2000 | 2000 | 2000 | 2000 | |||
3 | 559,304 | 556,813 | 556,378 | 557,014 | 556,610 | ||
559,489.8 | 557,463.4 | - | 557,456.3 | 557,204.5 | |||
- | - | - | 339.73 | 328.36 | |||
2000 | 2000 | 2000 | 2000 | 2000 |
Index | Results | ITS [3] | HSII [18] | TPS [19] | TPS * | CMA | |
---|---|---|---|---|---|---|---|
150 | 1 | 206,721 | 206,925 | 206,721 | 206,721 | 206,721 | |
207,004.9 | 207,089.4 | - | 206,850 | 206,795.1 | |||
- | - | - | 123.08 | 98.09 | |||
400 | 400 | 400 | 400 | 400 | |||
2 | 206,761 | 207,102 | 206,761 | 206,761 | 206,761 | ||
207,153.6 | 207,280.5 | - | 206,889.9 | 206,853.9 | |||
- | - | - | 105.8 | 83.43 | |||
400 | 400 | 400 | 400 | 400 | |||
3 | 206,802 | 206,781 | 206,759 | 206,759 | 206,759 | ||
206,959.6 | 206,954.2 | - | 206,847.5 | 206,827.1 | |||
- | - | - | 87.49 | 59.3 | |||
400 | 400 | 400 | 400 | 400 | |||
200 | 1 | 370,137 | 370,265 | 369,851 | 369,851 | 369,840 * | |
370,533.3 | 370,530.5 | - | 370,111.8 | 370,032.1 | |||
- | - | - | 175.59 | 138.84 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
2 | 370,028 | 369,982 | 369,803 | 369,835 | 369,754 * | ||
370,351 | 370,183.6 | - | 370,125.6 | 370,058.2 | |||
- | - | - | 201.3 | 143.47 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
3 | 370,046 | 370,045 | 369,794 | 369,809 | 369,705 * | ||
370,390.7 | 370,345.8 | - | 370,080 | 369,918.3 | |||
- | - | - | 189.65 | 147.91 | |||
1200 | 1200 | 1200 | 1200 | 1200 | |||
250 | 1 | 582,282 | 581,819 | 581,671 | 581,543 | 581,439 * | |
583,069.4 | 582,283.7 | - | 581,933.2 | 581,777.2 | |||
- | - | - | 236.16 | 165.84 | |||
2000 | 2000 | 2000 | 2000 | 2000 | |||
2 | 582,145 | 581,691 | 581,492 | 581,463 | 581,364 * | ||
582,872.9 | 582,013.3 | - | 581,886.6 | 581,752.1 | |||
- | - | - | 270.95 | 204.35 | |||
2000 | 2000 | 2000 | 2000 | 2000 | |||
3 | 582,708 | 581,854 | 581,573 | 581,449 | 581,207 * | ||
583,525.8 | 582,590.8 | - | 581,952.3 | 581,837.4 | |||
- | - | - | 309.75 | 248.84 | |||
2000 | 2000 | 2000 | 2000 | 2000 |
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Zhang, S.; Mao, J.; Wang, N.; Li, D.; Ju, C. A Clustering-Enhanced Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem. Entropy 2023, 25, 87. https://doi.org/10.3390/e25010087
Zhang S, Mao J, Wang N, Li D, Ju C. A Clustering-Enhanced Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem. Entropy. 2023; 25(1):87. https://doi.org/10.3390/e25010087
Chicago/Turabian StyleZhang, Shufan, Jianlin Mao, Niya Wang, Dayan Li, and Chengan Ju. 2023. "A Clustering-Enhanced Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem" Entropy 25, no. 1: 87. https://doi.org/10.3390/e25010087
APA StyleZhang, S., Mao, J., Wang, N., Li, D., & Ju, C. (2023). A Clustering-Enhanced Memetic Algorithm for the Quadratic Minimum Spanning Tree Problem. Entropy, 25(1), 87. https://doi.org/10.3390/e25010087