In this paper, we consider the NP-hard problem of finding the minimum dominant resolving set of graphs. A vertex set

The primary metric dimension of graphs was just recently introduced in [_{1},_{2},_{k}} ⊆

is different for each

_{7} is given in _{2}_{3,}_{4}_{5}_{6}} is a minimal resolving set but not a dominating set of _{7} since _{7} is not adjacent to vertices in _{1}_{2}_{3,}_{4}_{5}_{6}} is a minimal dominant resolving set of _{7}. Thus, _{7}) = 5 and _{7}) = 6.

Both the metric dimension problem and the problem with the dominant set are NP-complete problems [

The metric dimension is determined theoretically for several graphs in the literature [

The dominant metric dimension is studied in [_{n}, cycle graph _{n}, star graph _{n}, complete graph _{n}, and complete bipartite graph _{m,n} are theoretically determined. It has been shown that _{n}), _{n}), _{n}), _{n}), _{n}), _{m,n}), _{n}, cycle graph _{n}, complete bipartite graph _{m,n}, complete graph _{n} and star graph _{n}. Also see more details in the literature [

The smallest dominating resolving set of graphs is being calculated heuristically for the first time in this study. To resolve the problem, we modify the operations of a binary version of the Archimedes optimization algorithm (BAOA). The theoretically generated graph results are used to test the proposed BAOA. On various graphs and theoretically generated graphs, the proposed algorithm is compared with competing algorithms.

The paper is organized as follows:

AOA is a physics-inspired algorithm, specifically Archimedes’ law. This algorithm, which is a member of the meta-heuristics class, was developed by Hashim et al. [_{best}).

During the AOA process, updates to density and volume change the acceleration based on the collision notion between objects. The general AOA steps are as follows:

where _{i} represents the ^{th} object among _{i} and _{i}, respectively, stand for the lower and upper bounds of the search space.

Use ^{th} object:

The acceleration (acc) of the ^{th} object is then initialized using the

In this stage, evaluate the starting population and select the object with the best fitness value. _{best}, _{best}, _{best}, and _{best} should be assigned.

where _{best} and _{best} represent the volume and density associated with the best item discovered so far.

_{c}), according to

The maximum number of iterations is _{c} grows exponentially over time until it reaches 1. _{s} in

where _{Best} denotes the acceleration of the optimal object _{Best}.

where

_{c} ≤ 0.5), ^{th} object in iteration _{c} > 0.5),

where _{1} equals 2.

where _{2} is equal to 6.

The parameter _{c}, i.e., δ = 2 × _{c}. The main role of this parameter is to maintain a proper balance between exploration and exploitation operations. The margin between the best object and the other object is higher during the first iterations, resulting in a high random walk. However, in the final iterations, the margin will be decreased and provide a low random walk.

where _{4}.

_{Best}, _{Best}, and _{Best} to identify the best objects.

Because it maintains a population of solutions and examines a vast area to find the best global solution, the Archimedes optimization algorithm can solve difficult optimization problems with numerous locally optimal solutions. This benefit enables the binary version of the algorithm to be applied to the dominant metric dimension problem. Using position vectors in the continuous real domain, objects can navigate the search space in the continuous version of AOA. By using an

The following equation is used in the initialization step:

where a rand is a random number between 0 and 1.

A transfer function is used to be able to map continuous values to binary ones. In this study, the sigmoid function (

where ^{d} indicates the continuous-valued position at dimension

The proposed algorithm deals with the dominant resolving set problem as an optimization problem where it searches for the best solution, so each object can be represented as a one-dimensional vector _{ij} = (_{i1}, _{i2}, _{ij}), _{ij} is a binary-valued position vector if the

Each solution in the population is represented by the algorithm as a string of binary values, where 1 indicates that the dominant resolving set will be chosen, in which case the corresponding value will be “1,” and 0 indicates that the dominant resolving set will not be chosen, in which case the corresponding value will be “0”.

Thus, the flowchart of the proposed BAOA algorithm is displayed in

In this section, the proposed BAOA is tested using graph results that are computed theoretically. The proposed BAOA is compared to the BWOA and BPSO on a complete graph, a star graph, a path graph, an alternate triangular snake with pendant edge graph, and an alternate quadrilateral snake graph.

The algorithm tests and comparisons were performed on a Windows 10 Ultimate 64-bit operating system; the processor was an Intel Core i7 running at 16 GB of RAM, the hard drive was 1TBHDD+1TBSSD, and the code was implemented in MATLAB 2021b. The parameter setting values are presented in

Algorithms | Parameter name | Value |
---|---|---|

Objects number | 30 | |

BAOA | Max iteration | 500 |

_{1} _{2} _{3} _{4} |
[2, 6, 2, 0.5] | |

Number of runs | 20 | |

Whales number | 30 | |

Decrease from 2 to 0 | ||

BWOA | _{2} |
Decrease from −1 to −2 |

Max iteration | 500 | |

Number of runs | 20 | |

Swarm size | 30 | |

_{1} |
Increases linearly from 0.5 to 2.5 | |

BPSO | _{2} |
Decreases linearly from 2.5 to 0.5 |

Inertia weight ( |
0.8 | |

Max iteration | 500 | |

Number of runs | 20 |

The BAOA, BWOA and BPSO have been run 20 times for each instance, and the results are summarized in

The first three columns contain the test instance name, the number of nodes and edges, respectively.

The fourth column contains the BAOA best solution (named BAOA _{best}) obtained in 20 runs;

The average execution time (

The sixth column contains the average number of generations for finishing BAOA _{best}.

The seven and the eighth column variance and standard deviation contain information on the average solution quality.

Instance | BAOA _{best} |
Iteration |
BWOA | Iteration | BPSO | Iteration | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{1} |
3 | 3 | 2 | 1 | 2 | 9.28 | 1 | 2 | 13.71 | 1 | |

_{2} |
4 | 6 | 3 | 1 | 3 | 32.65 | 4 | 3 | 29.28 | 3 | |

_{3} |
5 | 10 | 4 | 1 | 4 | 78.92 | 8 | 4 | 42.13 | 17 | |

_{4} |
6 | 15 | 5 | 73.68 | 3 | 5 | 95.24 | 9 | 5 | 58.16 | 5 |

_{5} |
7 | 21 | 6 | 104.21 | 5 | 6 | 147.83 | 13 | 6 | 93.04 | 22 |

_{6} |
8 | 28 | 7 | 8 | 7 | 192.39 | 5 | 7 | 184.21 | 14 | |

_{7} |
9 | 36 | 8 | 2 | 8 | 209.45 | 21 | 8 | 256.09 | 39 | |

_{8} |
10 | 45 | 9 | 316.39 | 9 | 9 | 15 | 9 | 338.05 | 7 | |

_{9} |
11 | 55 | 10 | 3 | 10 | 525.32 | 18 | 10 | 480.19 | 125 | |

_{10} |
12 | 66 | 11 | 7 | 11 | 742.99 | 141 | 11 | 623.53 | 10 | |

_{11} |
13 | 78 | 12 | 13 | 12 | 881.13 | 27 | 12 | 735.28 | 21 | |

_{12} |
14 | 91 | 13 | 6 | 13 | 1022.19 | 9 | 13 | 913.11 | 14 | |

_{13} |
15 | 105 | 14 | 8 | 14 | 1273.44 | 31 | 14 | 1319.72 | 29 | |

_{14} |
16 | 120 | 15 | 12 | 15 | 1379.15 | 116 | 15 | 1468.03 | 134 | |

_{15} |
17 | 136 | 16 | 9 | 16 | 1428.23 | 54 | 16 | 1704.09 | 22 | |

_{16} |
18 | 153 | 17 | 11 | 17 | 1592.71 | 12 | 17 | 1811.14 | 17 | |

_{17} |
19 | 171 | 18 | 7 | 18 | 1714.83 | 37 | 18 | 1525.54 | 26 | |

_{18} |
20 | 190 | 19 | 21 | 19 | 1995.03 | 45 | 19 | 1761.28 | 32 | |

_{19} |
21 | 210 | 20 | 19 | 20 | 2168.25 | 132 | 20 | 2056.87 | 71 | |

_{20} |
22 | 231 | 21 | 14 | 21 | 2409.12 | 96 | 21 | 2238.05 | 57 |

Instance | BAOA _{best} |
t (s) | Iteration | BWOA | Iteration | BPSO | Iteration | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{1} |
3 | 2 | 2 | 1 | 2 | 7.54 | 1 | 2 | 8.19 | 1 | |

_{2} |
4 | 3 | 3 | 1 | 3 | 34.02 | 7 | 3 | 31.4 | 2 | |

_{3} |
5 | 4 | 4 | 3 | 4 | 81.95 | 3 | 4 | 72.79 | 5 | |

_{4.} |
6 | 5 | 5 | 127.39 | 2 | 5 | 9 | 5 | 146.71 | 11 | |

_{5} |
7 | 6 | 6 | 9 | 6 | 239.36 | 17 | 6 | 267.25 | 8 | |

_{6} |
8 | 7 | 7 | 1 | 7 | 418.14 | 22 | 7 | 429.41 | 15 | |

_{7} |
9 | 8 | 8 | 4 | 8 | 547.58 | 36 | 8 | 514.99 | 27 | |

_{8} |
10 | 9 | 9 | 2 | 9 | 623.72 | 14 | 9 | 602.15 | 31 | |

_{9} |
11 | 10 | 10 | 3 | 10 | 735.91 | 8 | 10 | 699.73 | 23 | |

_{10} |
12 | 11 | 11 | 762.46 | 3 | 11 | 870.73 | 27 | 11 | 109 | |

_{11} |
13 | 12 | 12 | 836.84 | 8 | 12 | 954.23 | 10 | 12 | 4 | |

_{12} |
14 | 13 | 13 | 6 | 13 | 1063.15 | 79 | 13 | 1196.16 | 21 | |

_{13} |
15 | 14 | 14 | 11 | 14 | 1198.82 | 126 | 14 | 1278.52 | 117 | |

_{14} |
16 | 15 | 15 | 9 | 15 | 1286.19 | 18 | 15 | 1193.04 | 28 | |

_{15} |
17 | 16 | 16 | 15 | 16 | 1335.08 | 22 | 16 | 1402.15 | 142 | |

_{16} |
18 | 17 | 17 | 13 | 17 | 1493.12 | 135 | 17 | 1519.82 | 32 | |

_{17} |
19 | 18 | 18 | 10 | 18 | 1586.31 | 28 | 18 | 1664.98 | 108 | |

_{18} |
20 | 19 | 19 | 17 | 19 | 1768.53 | 151 | 19 | 1575.34 | 83 | |

_{19} |
21 | 20 | 20 | 14 | 20 | 1913.22 | 87 | 20 | 1789.61 | 45 | |

_{20} |
22 | 21 | 21 | 15 | 21 | 2108.38 | 52 | 21 | 1902.27 | 36 |

Instance | BAOA _{best} |
Iteration | BWOA | Iteration | BPSO | Iteration | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{3} |
3 | 2 | 2 | 1 | 2 | 2.54 | 1 | 2 | 4.31 | 1 | |

_{4} |
4 | 3 | 2 | 1 | 2 | 7.89 | 4 | 2 | 5.76 | 8 | |

_{5} |
5 | 4 | 2 | 14.45 | 1 | 2 | 23.92 | 5 | 2 | 3 | |

_{6} |
6 | 5 | 2 | 3 | 2 | 72.94 | 13 | 2 | 80.65 | 7 | |

_{7} |
7 | 6 | 3 | 1 | 3 | 95.78 | 9 | 3 | 109.12 | 11 | |

_{8} |
8 | 7 | 3 | 4 | 3 | 186.15 | 25 | 3 | 128.47 | 6 | |

_{9} |
9 | 8 | 3 | 12 | 3 | 268.29 | 18 | 3 | 192.71 | 27 | |

_{10} |
10 | 9 | 3 | 7 | 3 | 311.05 | 30 | 3 | 275.64 | 19 | |

_{11} |
11 | 10 | 4 | 15 | 4 | 356.48 | 21 | 4 | 334.92 | 123 | |

_{12} |
12 | 11 | 4 | 413.47 | 9 | 4 | 474.21 | 116 | 4 | 75 | |

_{13} |
13 | 12 | 4 | 3 | 4 | 591.09 | 27 | 4 | 618.41 | 38 | |

_{14} |
14 | 13 | 5 | 17 | 5 | 616.37 | 41 | 5 | 761.09 | 20 | |

_{15} |
15 | 14 | 5 | 8 | 5 | 894.12 | 135 | 5 | 882.56 | 113 | |

_{16} |
16 | 15 | 5 | 14 | 5 | 1025.81 | 22 | 5 | 907.18 | 29 | |

_{17} |
17 | 16 | 6 | 10 | 6 | 1136.45 | 117 | 6 | 1198.07 | 12 | |

_{18} |
18 | 17 | 6 | 4 | 6 | 1417.93 | 29 | 6 | 1576.51 | 8 | |

_{19} |
19 | 18 | 6 | 9 | 6 | 1641.18 | 26 | 6 | 1449.27 | 17 | |

_{20} |
20 | 19 | 7 | 2 | 7 | 1918.04 | 18 | 7 | 1685.93 | 24 |

Instance | BAOA _{best} |
Iteration | BWOA | Iteration | BPSO | Iteration | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{1}) |
5 | 5 | 2 | 1 | 2 | 55.27 | 4 | 2 | 76.12 | 2 | |

_{2} |
8 | 9 | 2 | 1 | 2 | 93.12 | 6 | 2 | 108.45 | 5 | |

_{3} |
11 | 13 | 3 | 135.78 | 4 | 3 | 5 | 3 | 157.81 | 9 | |

_{4} |
14 | 17 | 5 | 3 | 5 | 241.35 | 17 | 5 | 212.38 | 12 | |

_{5} |
17 | 21 | 6 | 7 | 6 | 349.02 | 9 | 6 | 375.95 | 27 | |

_{6} |
20 | 25 | 7 | 501.14 | 9 | 7 | 16 | 7 | 487.34 | 10 | |

_{7} |
23 | 29 | 8 | 5 | 8 | 694.62 | 29 | 8 | 643.29 | 22 | |

_{8} |
26 | 33 | 9 | 8 | 9 | 908.53 | 34 | 9 | 871.43 | 15 | |

_{9} |
29 | 37 | 10 | 1017.42 | 10 | 10 | 1213.44 | 18 | 10 | 42 | |

_{10} |
32 | 41 | 11 | 1182.61 | 10 | 11 | 1396.18 | 27 | 11 | 19 | |

_{11} |
35 | 45 | 12 | 15 | 12 | 1432.32 | 44 | 12 | 1327.91 | 32 | |

_{12} |
38 | 49 | 13 | 3 | 13 | 1547.49 | 19 | 13 | 1466.80 | 11 | |

_{13} |
41 | 53 | 14 | 11 | 14 | 1714.18 | 131 | 14 | 1534.11 | 17 | |

_{14} |
44 | 57 | 15 | 14 | 15 | 1848.71 | 96 | 15 | 1675.86 | 35 | |

_{15} |
47 | 61 | 16 | 12 | 16 | 1954.35 | 17 | 16 | 1811.99 | 122 | |

_{16} |
50 | 65 | 17 | 18 | 17 | 2078.11 | 39 | 17 | 1927.52 | 18 | |

_{17} |
53 | 69 | 18 | 11 | 18 | 2215.46 | 42 | 18 | 2083.28 | 29 | |

_{18} |
56 | 73 | 19 | 16 | 19 | 2399.28 | 29 | 19 | 2204.39 | 24 | |

_{19} |
59 | 77 | 20 | 2 | 20 | 2523.13 | 34 | 20 | 2391.15 | 141 | |

_{20} |
62 | 81 | 21 | 11 | 21 | 2796.17 | 22 | 21 | 2503.57 | 27 | |

_{21} |
65 | 85 | 22 | 19 | 22 | 2981.43 | 153 | 22 | 2655.29 | 36 | |

_{22} |
68 | 89 | 23 | 26 | 23 | 3149.24 | 40 | 24 | 2810.92 | 48 | |

_{23} |
71 | 93 | 24 | 19 | 24 | 3211.15 | 25 | 24 | 2673.42 | 21 | |

_{24} |
74 | 97 | 25 | 1 | 25 | 3319.28 | 113 | 25 | 2716.19 | 19 | |

_{25} |
77 | 101 | 26 | 17 | 26 | 3584.93 | 21 | 27 | 2931.43 | 32 | |

_{26} |
80 | 105 | 27 | 19 | 27 | 3706.21 | 32 | 28 | 2815.95 | 120 | |

_{27} |
83 | 109 | 28 | 19 | 29 | 3851.75 | 68 | 28 | 3099.22 | 29 | |

_{28} |
86 | 113 | 29 | 19 | 30 | 3638.44 | 45 | 31 | 3186.36 | 23 | |

_{29} |
89 | 117 | 30 | 19 | 31 | 3992.65 | 36 | 32 | 3319.74 | 31 | |

_{30} |
92 | 121 | 31 | 19 | 34 | 4065.22 | 30 | 32 | 3479.15 | 22 | |

_{31} |
95 | 125 | 32 | 18 | 35 | 4198.13 | 42 | 33 | 3611.89 | 38 | |

_{32} |
98 | 129 | 33 | 20 | 36 | 4282.57 | 33 | 35 | 3843.54 | 25 | |

_{33} |
101 | 133 | 34 | 2 | 37 | 4419.16 | 19 | 36 | 3954.93 | 16 |

Our stopping criterion is the cardinality of the dominant resolving set that reaches the known dominant metric dimension of the complete graph. BAOA takes 168.76 s on _{6}, and it takes 8 iterations to complete BAOA to achieve the best solution.

Regarding BAOA results, _{k},1 ≤ _{3}, the time needed for BAOA is 69.01 s and reaches the best solution after 3 iteration. In [

Regarding BAOA results, _{n}, 3 ≤ _{6}, the time required for BAOA is 48.86 s, with 3 iterations required to achieve the best solution.

BAOA found an optimal solution for the alternate triangular snake with pendant edge _{k}), _{4}

Regarding BAOA results, _{k}), _{3}), the time needed for BAOA is 205.23 s and reaches the best solution after 2 iterations.

Instance | BAOA _{best} |
t (s) | Iteration | BWOA | Iteration | BPSO | Iteration | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

_{1}) |
5 | 5 | 3 | 1 | 3 | 108.63 | 1 | 3 | 95.12 | 1 | |

_{2}) |
9 | 10 | 4 | 154.17 | 1 | 4 | 191.27 | 6 | 4 | 3 | |

_{3}) |
13 | 15 | 5 | 2 | 5 | 287.15 | 4 | 5 | 292.36 | 9 | |

_{4}) |
17 | 20 | 7 | 5 | 7 | 561.12 | 9 | 7 | 523.49 | 21 | |

_{5}) |
21 | 25 | 8 | 7 | 643.38 | 22 | 7 | 678.22 | 14 | ||

_{6}) |
25 | 30 | 8 | 17 | 8 | 801.79 | 35 | 8 | 835.69 | 18 | |

_{7}) |
29 | 35 | 9 | 44 | 9 | 982.46 | 19 | 9 | 923.15 | 30 | |

_{8}) |
33 | 40 | 10 | 803.59 | 18 | 10 | 52 | 10 | 1174.38 | 23 | |

_{9}) |
37 | 45 | 11 | 36 | 11 | 1276.84 | 43 | 11 | 1305.47 | 19 | |

_{10}) |
41 | 50 | 12 | 25 | 12 | 1421.75 | 28 | 12 | 1398.71 | 6 | |

_{11}) |
45 | 55 | 13 | 51 | 13 | 1538.52 | 152 | 13 | 1512.87 | 35 | |

_{12}) |
49 | 60 | 14 | 37 | 14 | 1673.21 | 63 | 14 | 1609.53 | 41 | |

_{13}) |
53 | 65 | 15 | 46 | 15 | 1825.43 | 49 | 15 | 1715.19 | 64 | |

_{14}) |
57 | 70 | 16 | 34 | 16 | 1913.92 | 38 | 16 | 1873.14 | 112 | |

_{15}) |
61 | 75 | 17 | 42 | 17 | 2056.27 | 106 | 17 | 2119.25 | 88 | |

_{16}) |
65 | 80 | 18 | 45 | 18 | 2172.49 | 47 | 18 | 2205.82 | 95 | |

_{17}) |
69 | 85 | 19 | 42 | 20 | 2328.72 | 134 | 19 | 2289.07 | 61 | |

_{18}) |
73 | 90 | 20 | 46 | 20 | 2416.41 | 52 | 20 | 2397.28 | 49 | |

_{19}) |
77 | 95 | 21 | 1 | 21 | 2673.12 | 29 | 21 | 2473.16 | 18 | |

_{20}) |
81 | 100 | 22 | 42 | 22 | 2768.95 | 46 | 22 | 2583.55 | 105 | |

_{21}) |
85 | 105 | 23 | 17 | 23 | 2995.64 | 37 | 23 | 2839.31 | 22 | |

_{22}) |
89 | 110 | 24 | 44 | 24 | 3082.18 | 55 | 24 | 2765.24 | 53 | |

_{23}) |
93 | 115 | 25 | 40 | 25 | 3176.27 | 61 | 25 | 2932.89 | 47 | |

_{24}) |
97 | 120 | 26 | 43 | 26 | 3287.34 | 42 | 26 | 3139.27 | 19 | |

_{25}) |
101 | 125 | 27 | 3 | 27 | 3375.16 | 74 | 27 | 3290.58 | 12 | |

_{26}) |
105 | 130 | 28 | 11 | 28 | 3592.28 | 25 | 28 | 3367.12 | 28 | |

_{27}) |
109 | 135 | 29 | 2 | 29 | 3787.12 | 36 | 29 | 3515.44 | 17 | |

_{28}) |
113 | 140 | 30 | 39 | 30 | 3971.47 | 58 | 30 | 3636.78 | 144 | |

_{29}) |
117 | 145 | 31 | 26 | 31 | 4218.38 | 22 | 31 | 3849.25 | 25 | |

_{30}) |
121 | 150 | 32 | 43 | 32 | 4416.22 | 29 | 32 | 3698.47 | 33 | |

_{31}) |
125 | 155 | 33 | 1 | 33 | 4594.19 | 84 | 33 | 3953.13 | 40 | |

_{32}) |
129 | 160 | 34 | 2 | 34 | 4683.85 | 23 | 34 | 4165.73 | 25 | |

_{33}) |
133 | 165 | 35 | 4 | 35 | 4925.03 | 49 | 35 | 4379.24 | 32 |

Experiments in this paper are performed on a subset of complete graph instances with

_{2} is 3 and reaches 9.11 s. _{7} is 3 and reaches 83.91 s. All figures show the superiority of the proposed BAOA.

In this paper, the operations of a binary version of the Archimedes optimization algorithm BAOA are adapted to solve the dominant metric dimension problem. The proposed BAOA is tested using graph results that are computed theoretically. The proposed algorithm is compared to competitive algorithms on graphs that are computed theoretically and other graphs. The performance of the proposed BAOA outperforms that of the BWOA and BPSO.

An Open Problem. Other efficient metaheuristic algorithms for determining any variant of metric dimension that does not compute the previous heuristically for any regular graph or planar graph, as well as comparing them to competitive algorithms.

This work was not supported by any funding.

The authors received no specific funding for this study.

Conceptualization, methodology, writing review and formal analysis, Basma Mohamed; investigation, resources, Linda Mohaisen; writing–original draught preparation, validation, editing and visualization, Mohammed Amin. All authors have read and agreed to the published version of the manuscript.

The data underlying the results presented in the study are available within the manuscript.

The authors declare that they have no conflicts of interest to report regarding the present study.

_{n}, lollipop graph

_{m,n}and petersen graph

_{n,m}