# Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. The Capacitated Vertex K-Center Problem

#### 2.1. Classical Integer Programming Formulation

#### 2.2. A New Formulation Based on the Minimum Capacitated Dominating Set

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. An Exact Algorithm for the Capacitated Vertex K-Center Problem

Algorithm 1: An exact algorithm for the capacitated vertex k-center problem |

**Theorem**

**3.**

**Proof.**

## 4. Empirical Performance Evaluation

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Running time and optimal solution size found by Algorithm 1 and the classical formulation F1C over Set 1 using Gurobi 9.0.0. Best running times are in bold.

Running Time (Seconds) | |||||
---|---|---|---|---|---|

Instance | $\mathit{n}$ | $\mathit{k}$ | OPT | Algorithm 1 | F1C |

kroA100_Q1 | 100 | 5 | 895.64 | 2.32 | 10.42 |

6 | 814.87 | 1.46 | 4.18 | ||

kroA100_Q2 | 100 | 10 | 606.57 | 1.86 | 12.64 |

11 | 554.04 | 1.43 | 9.07 | ||

kroA100_Q3 | 100 | 20 | 411.61 | 4.95 | 32.45 |

21 | 376.96 | 2.65 | 33.78 | ||

kroA100_Q4 | 100 | 40 | 325.04 | 1.00 | 12.90 |

41 | 314.23 | 0.72 | 2.67 | ||

kroB100_Q1 | 100 | 5 | 924.57 | 3.39 | 9.27 |

6 | 823.66 | 1.23 | 5.43 | ||

kroB100_Q2 | 100 | 10 | 602.9 | 2.25 | 7.44 |

11 | 559.35 | 1.34 | 4.99 | ||

kroB100_Q3 | 100 | 20 | 425.73 | 11.91 | 26.30 |

21 | 414.77 | 6.56 | 19.76 | ||

kroB100_Q4 | 100 | 40 | 343.57 | 0.77 | 2.22 |

41 | 330.84 | 0.75 | 2.69 | ||

kroC100_Q1 | 100 | 5 | 867.16 | 1.79 | 6.61 |

6 | 762.27 | 1.07 | 8.94 | ||

kroC100_Q2 | 100 | 10 | 580.53 | 2.69 | 3.68 |

11 | 545.69 | 2.27 | 3.32 | ||

kroC100_Q3 | 100 | 20 | 426.82 | 29.29 | 38.61 |

21 | 415.32 | 13.89 | 124.56 | ||

kroC100_Q4 | 100 | 40 | 307.65 | 0.90 | 2.50 |

41 | 288.53 | 0.65 | 1.39 | ||

eil101_Q1 | 101 | 5 | 21.21 | 4.44 | 10.40 |

6 | 18.68 | 1.62 | 9.03 | ||

eil101_Q2 | 101 | 10 | 15.23 | 17.63 | 25.27 |

11 | 13.92 | 6.96 | 18.20 | ||

eil101_Q3 | 101 | 20 | 10.44 | 10.66 | 9.04 |

21 | 10.29 | 7.51 | 5.15 | ||

eil101_Q4 | 101 | 40 | 8.6 | 4.94 | 1.59 |

41 | 8.48 | 2.32 | 1.89 | ||

lin105_Q1 | 105 | 5 | 677.44 | 6.14 | 7.31 |

6 | 610.45 | 2.81 | 6.69 | ||

lin105_Q2 | 105 | 10 | 555.0 | 99.23 | 27.35 |

11 | 476.05 | 152.56 | 27.1 | ||

lin105_Q3 | 105 | 20 | 307.0 | 3.51 | 15.74 |

21 | 307.0 | 3.35 | 25.69 | ||

lin105_Q4 | 105 | 40 | 177.01 | 1.31 | 2.09 |

41 | 162.69 | 0.84 | 1.62 | ||

pr107_Q1 | 107 | 5 | 2630.58 | 21.13 | 868.16 |

6 | 1068.87 | 1.58 | 2.31 | ||

pr107_Q2 | 107 | 10 | 894.42 | 3.07 | 8.51 |

11 | 824.62 | 2.79 | 2.75 | ||

pr107_Q3 | 107 | 20 | 538.51 | 3.17 | 1.55 |

21 | 447.21 | 3.04 | 2.48 | ||

pr107_Q4 | 107 | 40 | 282.84 | 1.88 | 1.10 |

41 | 282.84 | 1.86 | 0.93 | ||

Average time | 9.61 | 30.58 |

**Table 2.**Running time and optimal solution size found by Algorithm 1 and the classical formulation F1C over Set 2 using Gurobi 9.0.0. Best running times are in bold.

Running Time (Seconds) | |||||
---|---|---|---|---|---|

Instance | $\mathit{n}$ | $\mathit{k}$ | OPT | Algorithm 1 | F1C |

kroa200_Q1 | 200 | 5 | 919.03 | 25.61 | 133.75 |

6 | 808.66 | 10.53 | 67.71 | ||

kroa200_Q2 | 200 | 10 | 599.47 | 34.48 | 247.57 |

11 | 569.94 | 41.1 | 443.19 | ||

kroa200_Q3 | 200 | 20 | 415.49 | 534.92 | 1568.01 |

21 | 403.1 | 824.89 | 852.34 | ||

kroa200_Q4 | 200 | 40 | 293.29 | 76.65 | 242.6 |

41 | 287.45 | 59.69 | 166.92 | ||

kroB200_Q1 | 200 | 5 | 897.66 | 17.59 | 83.57 |

6 | 784.18 | 9.22 | 52.55 | ||

kroB200_Q2 | 200 | 10 | 589.86 | 41.29 | 904.5 |

11 | 567.5 | 27.26 | 126.71 | ||

kroB200_Q3 | 200 | 20 | 412.14 | 1352.51 | 3903.28 |

21 | 399.48 | 391.72 | 1230.63 | ||

kroB200_Q4 | 200 | 40 | 289.27 | 483.67 | 37,907.53 |

41 | 282.4 | 69.25 | 35,200.62 | ||

ts225_Q1 | 225 | 5 | 4000.0 | 127.16 | 118.97 |

6 | 3605.55 | 44.26 | 108.12 | ||

ts225_Q2 | 225 | 10 | 3041.38 | 840.59 | 3037.34 |

11 | 3000.0 | 176.57 | 4372.96 | ||

ts225_Q3 | 225 | 20 | 2000.0 | 407.86 | 473.29 |

21 | 1802.77 | 224.0 | 1566.91 | ||

ts225_Q4 | 225 | 40 | 1414.21 | 194.32 | 1115.63 |

41 | 1118.03 | 170.12 | 4623.0 | ||

pr226_Q1 | 226 | 5 | 4172.52 | 140.19 | 481.23 |

6 | 3778.97 | 53.08 | 213.13 | ||

pr226_Q2 | 226 | 10 | 2863.56 | 102.9 | >86,400 |

11 | 2844.29 | 73.39 | >86,400 | ||

pr226_Q3 | 226 | 20 | 2450.51 | 410.74 | >86,400 |

21 | 2300.54 | 186.73 | >86,400 | ||

pr226_Q4 | 226 | 40 | 1320.98 | 162.73 | >86,400 |

41 | 1166.19 | 87.73 | >86,400 | ||

gr229_Q1 | 229 | 5 | 50.26 | 15971.84 | 3036.74 |

6 | 37.94 | 120.27 | 192.09 | ||

gr229_Q2 | 229 | 10 | 37.94 | 9548.37 | 746.46 |

11 | 28.84 | 4216.9 | 685.63 | ||

gr229_Q3 | 229 | 20 | 23.23 | 8412.5 | 1129.62 |

21 | 22.61 | 2091.64 | 688.22 | ||

gr229_Q4 | 229 | 40 | 19.78 | 4738.61 | 1135.16 |

41 | 19.67 | 7827.95 | 745.42 | ||

a280_Q1 | 280 | 5 | 69.85 | 734.93 | 626.56 |

6 | 58.24 | 50.37 | 340.37 | ||

a280_Q2 | 280 | 10 | 45.25 | 93.96 | 333.76 |

11 | 42.52 | 66.67 | 328.59 | ||

a280_Q3 | 280 | 20 | 31.24 | 700.75 | 6289.73 |

21 | 28.84 | 376.45 | 2186.42 | ||

a280_Q4 | 280 | 40 | 21.54 | 846.97 | 1046.82 |

41 | 20.39 | 772.48 | 21,710.55 | ||

Average time | 1332.78 | >13,726.34 |

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**MDPI and ACS Style**

Cornejo Acosta, J.A.; García Díaz, J.; Menchaca-Méndez, R.; Menchaca-Méndez, R.
Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem. *Mathematics* **2020**, *8*, 1551.
https://doi.org/10.3390/math8091551

**AMA Style**

Cornejo Acosta JA, García Díaz J, Menchaca-Méndez R, Menchaca-Méndez R.
Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem. *Mathematics*. 2020; 8(9):1551.
https://doi.org/10.3390/math8091551

**Chicago/Turabian Style**

Cornejo Acosta, José Alejandro, Jesús García Díaz, Ricardo Menchaca-Méndez, and Rolando Menchaca-Méndez.
2020. "Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem" *Mathematics* 8, no. 9: 1551.
https://doi.org/10.3390/math8091551