Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems
Abstract
:1. Introduction
2. VNS Algorithm
- Observation 1: A local minimum for one neighborhood structure is not necessarily so for another;
- Observation 2: A global minimum is a local minimum for all of the possible neighborhood structures;
- Observation 3: For many problems, local minimums for one or several neighborhoods are relatively similar to each other.
3. Proposed Neighborhood Structure
4. Benchmarking
4.1. Test on QUBO Problems
4.2. Test on Max-Cut Problems
- 1.
- if , then ;
- 2.
- if , then .
4.3. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Problem Number | n | Best Known | VNS | B-VNS | Test (p-Value) * | |||||
---|---|---|---|---|---|---|---|---|---|---|
BestDif | AvgDif | Time ** | BestDif | AvgDif | Time ** | Dif | Time | |||
1a | 50 | 3414 | 0 | 1.667 | 0.003 | 0 | 1 | 0.003 | 0.459 | *** |
2a | 60 | 6063 | 0 | 0 | 0.012 | 0 | 0 | 0.011 | - | 0.006 |
3a | 70 | 6037 | 0 | 8.9 | 0.017 | 0 | 11.467 | 0.016 | 0.773 | *** |
4a | 80 | 8598 | 0 | 0 | 0.035 | 0 | 0 | 0.030 | - | 0.009 |
5a | 50 | 5737 | 0 | 0 | 0.004 | 0 | 3.867 | 0.003 | - | 0.041 |
6a | 30 | 3980 | 0 | 0 | *** | 0 | 0 | *** | - | *** |
7a | 30 | 4541 | 0 | 0 | *** | 0 | 0 | *** | - | *** |
8a | 100 | 11,109 | 0 | 1.467 | 0.128 | 0 | 0 | 0.121 | - | *** |
1b | 40 | 133 | 0 | 18.033 | *** | 0 | 21 | *** | 0.512 | - |
2b | 50 | 121 | 0 | 0.733 | *** | 0 | 2.667 | *** | 0.096 | *** |
3b | 60 | 118 | 0 | 4.667 | 0.001 | 0 | 8.533 | 0.001 | 0.065 | 0.401 |
4b | 70 | 129 | 0 | 13.867 | 0.004 | 0 | 18.733 | 0.003 | 0.112 | 0.005 |
5b | 80 | 150 | 0 | 0 | 0.012 | 0 | 0 | 0.011 | - | *** |
6b | 90 | 146 | 13 | 35.933 | 0.021 | 13 | 37.833 | 0.016 | 0.277 | *** |
7b | 80 | 160 | 0 | 0 | 0.027 | 0 | 2.533 | 0.025 | - | *** |
8b | 90 | 145 | 0 | 6.633 | 0.062 | 0 | 5.433 | 0.053 | 0.307 | *** |
9b | 100 | 137 | 0 | 2 | 0.156 | 0 | 2.433 | 0.141 | 1 | *** |
10b | 125 | 154 | 0 | 0.233 | 0.467 | 0 | 0.233 | 0.414 | 1 | *** |
1c | 40 | 5058 | 0 | 0 | 0.001 | 0 | 0 | 0.001 | - | 0.507 |
2c | 50 | 6213 | 0 | 0 | 0.003 | 0 | 0 | 0.003 | - | 0.107 |
3c | 60 | 6665 | 0 | 0 | 0.015 | 0 | 0 | 0.013 | - | 0.003 |
4c | 70 | 7398 | 0 | 0 | 0.020 | 0 | 0 | 0.017 | - | *** |
5c | 80 | 7362 | 0 | 0.867 | 0.035 | 0 | 0 | 0.028 | - | *** |
6c | 90 | 5824 | 0 | 27.467 | 0.048 | 0 | 21.167 | 0.047 | 0.186 | 0.043 |
7c | 100 | 7225 | 0 | 0 | 0.134 | 0 | 0 | 0.123 | - | *** |
1d | 100 | 6333 | 0 | 16.9 | 0.135 | 0 | 13.733 | 0.129 | 0.583 | 0.006 |
2d | 100 | 6579 | 0 | 31.967 | 0.152 | 0 | 19.633 | 0.145 | 0.214 | 0.022 |
3d | 100 | 9261 | 0 | 14.567 | 0.157 | 0 | 16.067 | 0.144 | 0.658 | *** |
4d | 100 | 10,727 | 0 | 5.367 | 0.166 | 0 | 9.067 | 0.151 | 0.056 | *** |
5d | 100 | 11,626 | 0 | 11.633 | 0.179 | 0 | 14.7 | 0.166 | 0.471 | 0.001 |
6d | 100 | 14,207 | 0 | 5 | 0.171 | 0 | 1.667 | 0.155 | 0.313 | *** |
7d | 100 | 14,476 | 0 | 8.9 | 0.194 | 0 | 7.733 | 0.173 | 0.763 | *** |
8d | 100 | 16,352 | 0 | 0 | 0.176 | 0 | 0 | 0.162 | - | *** |
9d | 100 | 15,656 | 0 | 1.13 | 0.180 | 0 | 0.3 | 0.164 | 0.305 | *** |
10d | 100 | 19,102 | 0 | 0 | 0.184 | 0 | 0 | 0.170 | - | *** |
1e | 200 | 16,464 | 0 | 12.833 | 4.689 | 0 | 11.767 | 4.237 | 0.576 | *** |
2e | 200 | 23,395 | 0 | 8 | 5.635 | 0 | 7.067 | 5.232 | 0.579 | 0.001 |
3e | 200 | 25,243 | 0 | 0 | 6.172 | 0 | 0 | 5.731 | - | *** |
4e | 200 | 35,594 | 0 | 0.533 | 5.071 | 0 | 0.533 | 4.697 | 1 | *** |
5e | 200 | 35,154 | 0 | 20.33 | 5.995 | 0 | 31.233 | 5.924 | 0.127 | 0.264 |
1f | 500 | 61,194 | 0 | 2 | 578.194 | 0 | 1.2 | 559.644 | 0.679 | 0.004 |
2f | 500 | 100,161 | 0 | 0.1 | 545.884 | 0 | 0.2 | 521.028 | 0.570 | *** |
3f | 500 | 138,035 | 0 | 38.967 | 521.415 | 0 | 37.9 | 521.502 | 0.594 | 0.971 |
4f | 500 | 172,771 | 0 | 33.6 | 440.721 | 0 | 18 | 450.550 | 0.354 | 0.050 |
5f | 500 | 190,507 | 0 | 2.833 | 499.021 | 0 | 3.3 | 511.853 | 0.513 | 0.04 |
n | Problem Number | Best Known | VNS | B-VNS | Test (p-Value) * | |||||
---|---|---|---|---|---|---|---|---|---|---|
BestDif | AvgDif | Time ** | BestDif | AvgDif | Time ** | Dif | Time | |||
50 | 1 | 2098 | 68 | 127.3 | 0.003 | 0 | 93.867 | 0.003 | 0.062 | 0.305 |
2 | 3702 | 0 | 15 | 0.003 | 0 | 22.967 | 0.003 | 0.497 | 0.006 | |
3 | 4626 | 0 | 11.367 | 0.003 | 0 | 19 | 0.003 | 0.248 | 0.025 | |
4 | 3544 | 0 | 21.533 | 0.003 | 0 | 19.733 | 0.003 | 0.863 | 0.006 | |
5 | 4012 | 0 | 10.667 | 0.003 | 0 | 2.933 | 0.003 | 0.170 | 0.677 | |
6 | 3693 | 0 | 1.933 | 0.003 | 0 | 2.9 | 0.003 | 0.654 | 0.190 | |
7 | 4520 | 0 | 4.6 | 0.003 | 0 | 4.867 | 0.003 | 0.288 | 0.031 | |
8 | 4216 | 0 | 18 | 0.003 | 0 | 7.333 | 0.003 | 0.117 | - | |
9 | 3780 | 0 | 19.367 | 0.005 | 0 | 20.267 | 0.003 | 0.887 | *** | |
10 | 3507 | 0 | 27.733 | 0.005 | 0 | 32.867 | 0.003 | 0.602 | *** | |
100 | 1 | 7970 | 42 | 150.867 | 0.080 | 0 | 173.133 | 0.076 | 0.163 | 0.034 |
2 | 11,036 | 0 | 15.333 | 0.083 | 0 | 19.333 | 0.078 | 0.732 | 0.001 | |
3 | 12,723 | 0 | 0 | 0.071 | 0 | 0 | 0.068 | - | 0.039 | |
4 | 10,368 | 0 | 8.333 | 0.078 | 0 | 11.533 | 0.072 | 0.984 | *** | |
5 | 9083 | 0 | 44.467 | 0.085 | 0 | 49.167 | 0.079 | 0.682 | 0.006 | |
6 | 10,210 | 0 | 2.067 | 0.088 | 0 | 4.767 | 0.080 | 0.910 | 0.006 | |
7 | 10,125 | 0 | 24.467 | 0.082 | 0 | 26.533 | 0.072 | 0.770 | *** | |
8 | 11,435 | 0 | 8.867 | 0.079 | 0 | 9 | 0.077 | 0.820 | 0.139 | |
9 | 11,455 | 0 | 0.6 | 0.081 | 0 | 0.6 | 0.075 | 1 | 0.004 | |
10 | 12,565 | 0 | 18.667 | 0.078 | 0 | 12.933 | 0.069 | 0.247 | *** | |
250 | 1 | 45,607 | 0 | 8 | 11.873 | 0 | 10.133 | 10.890 | 0.458 | *** |
2 | 44,810 | 0 | 59.267 | 12.123 | 0 | 45.033 | 11.429 | 0.147 | 0.005 | |
3 | 49,037 | 0 | 0 | 8.996 | 0 | 0 | 8.662 | - | 0.045 | |
4 | 41,274 | 0 | 20.6 | 10.539 | 0 | 33.133 | 9.743 | 0.067 | *** | |
5 | 47,961 | 0 | 15.933 | 9.672 | 0 | 10.933 | 8.808 | 0.611 | *** | |
6 | 41,014 | 0 | 8.6 | 11.766 | 0 | 11.5 | 10.973 | 0.876 | *** | |
7 | 46,757 | 0 | 0 | 10.624 | 0 | 0 | 10.191 | - | 0.050 | |
8 | 35,726 | 52 | 214.200 | 13.311 | 0 | 177 | 12.196 | 0.297 | *** | |
9 | 48,916 | 0 | 23.100 | 11.330 | 0 | 27.233 | 10.341 | 0.433 | *** | |
10 | 40,442 | 0 | 3.533 | 12.526 | 0 | 2.2 | 11.211 | 0.688 | *** | |
500 | 1 | 116,586 | 0 | 5.333 | 586.406 | 0 | 6.267 | 592.798 | 0.677 | 0.398 |
2 | 128,339 | 0 | 2.5 | 459.926 | 0 | 4.4 | 455.013 | 0.402 | 0.374 | |
3 | 130,812 | 0 | 0 | 501.773 | 0 | 0 | 496.806 | - | 0.432 | |
4 | 130,097 | 0 | 28.933 | 518.133 | 0 | 26.733 | 523.351 | 0.486 | 0.321 | |
5 | 125,487 | 0 | 10.4 | 521.381 | 0 | 2.6 | 516.252 | 0.380 | 0.300 |
Graph | Problem | n | Best Known | VNS | B-VNS | Test (p-Value) * | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
BestDif | AvgDif | Time ** | BestDif | AvgDif | Time ** | Dif | Time | ||||
Random | G1 | 800 | 11624 | 0 | 0.033 | 180.13 | 0 | 2.533 | 158.702 | *** | *** |
G2 | 800 | 11620 | 0 | 7.133 | 185.804 | 0 | 6.8 | 162.854 | 0.830 | *** | |
G3 | 800 | 11622 | 0 | 1.733 | 195.169 | 0 | 2.833 | 172.003 | 0.222 | *** | |
G4 | 800 | 11646 | 0 | 0.567 | 193.595 | 0 | 0.633 | 175.664 | 0.507 | *** | |
G5 | 800 | 11631 | 0 | 5.133 | 191.32 | 0 | 4.433 | 169.104 | 0.654 | *** | |
Random () | G6 | 800 | 2178 | 0 | 1.867 | 203.065 | 0 | 2.2 | 136.156 | 0.299 | *** |
G7 | 800 | 2006 | 0 | 4.533 | 195.770 | 0 | 5.2 | 169.933 | 0.286 | *** | |
G8 | 800 | 2005 | 0 | 3.4 | 154.650 | 0 | 2.733 | 164.133 | 0.845 | *** | |
G9 | 800 | 2054 | 0 | 4.3 | 195.341 | 1 | 4.6 | 169.740 | 0.622 | *** | |
G10 | 800 | 2000 | 0 | 3.2 | 160.173 | 0 | 2.333 | 141.589 | 0.191 | *** | |
Toroidal | G11 | 800 | 564 | 14 | 25.267 | 66.274 | 20 | 27.733 | 44.726 | 0.001 | *** |
G12 | 800 | 556 | 18 | 24.4 | 66.716 | 16 | 24.133 | 57.920 | 0.916 | *** | |
G13 | 800 | 582 | 16 | 22.8 | 68.079 | 18 | 24.067 | 62.023 | 0.127 | *** | |
Planar | G14 | 800 | 3064 | 29 | 37.733 | 78.656 | 32 | 39.6 | 69.983 | 0.087 | *** |
G15 | 800 | 3050 | 31 | 38.633 | 77.195 | 32 | 39.867 | 67.350 | 0.179 | *** | |
Random | G43 | 1000 | 6660 | 1 | 8.967 | 431.314 | 1 | 9.733 | 381.073 | 0.323 | *** |
G44 | 1000 | 6650 | 3 | 8.467 | 428.178 | 2 | 9.533 | 375.822 | 0.114 | *** | |
G45 | 1000 | 6654 | 0 | 12.067 | 425.304 | 1 | 11.033 | 374.124 | 0.445 | *** | |
G46 | 1000 | 6654 | 9 | 16.1 | 410.047 | 5 | 16.633 | 373.864 | 0.494 | *** | |
G47 | 1000 | 6654 | 9 | 20.433 | 415.607 | 13 | 20.267 | 370.332 | 0.472 | *** | |
Planar | G51 | 1000 | 3846 | 39 | 47.433 | 194.891 | 39 | 49.533 | 192.633 | 0.070 | 0.007 |
G52 | 1000 | 3849 | 41 | 49.1 | 126.208 | 42 | 50.033 | 110.035 | 0.265 | *** |
Problem | n | Best Known | VNS | B-VNS | Test (p-Value) * | |||||
---|---|---|---|---|---|---|---|---|---|---|
BestDif | AvgDif | Time ** | BestDif | AvgDif | Time ** | Dif | Time | |||
sg3dl052000 | 125 | 112 | 0 | 1.2 | 0.042 | 0 | 1.8 | 0.039 | 0.058 | *** |
sg3dl054000 | 125 | 114 | 0 | 1.333 | 0.043 | 0 | 2.133 | 0.039 | 0.158 | *** |
sg3dl056000 | 125 | 110 | 0 | 1.333 | 0.041 | 0 | 1.467 | 0.038 | 0.803 | *** |
sg3dl058000 | 125 | 108 | 0 | 1.333 | 0.043 | 0 | 1.6 | 0.039 | 0.312 | *** |
sg3dl0510000 | 125 | 112 | 0 | 3.267 | 0.042 | 0 | 2.4 | 0.039 | 0.030 | *** |
sg3dl102000 | 1000 | 900 | 20 | 28.867 | 402.421 | 18 | 30.200 | 350.943 | 0.237 | *** |
sg3dl104000 | 1000 | 896 | 18 | 28.667 | 431.393 | 20 | 28.733 | 351.443 | 0.928 | *** |
sg3dl106000 | 1000 | 886 | 24 | 31.267 | 359.999 | 28 | 34.467 | 322.743 | 0.004 | *** |
sg3dl108000 | 1000 | 880 | 16 | 26.867 | 380.613 | 20 | 28.600 | 311.638 | 0.058 | *** |
sg3dl1010000 | 1000 | 890 | 18 | 28.800 | 390.797 | 20 | 29.733 | 354.691 | 0.397 | *** |
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Pambudi, D.; Kawamura, M. Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems. Algorithms 2022, 15, 192. https://doi.org/10.3390/a15060192
Pambudi D, Kawamura M. Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems. Algorithms. 2022; 15(6):192. https://doi.org/10.3390/a15060192
Chicago/Turabian StylePambudi, Dhidhi, and Masaki Kawamura. 2022. "Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems" Algorithms 15, no. 6: 192. https://doi.org/10.3390/a15060192
APA StylePambudi, D., & Kawamura, M. (2022). Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems. Algorithms, 15(6), 192. https://doi.org/10.3390/a15060192