# Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. TSP and SmTSP

#### 2.2. mTSP from 1960 to 1975

#### 2.3. mTSP from 1976 to 1995

#### 2.4. mTSP from 1996 to 2005

#### 2.5. mTSP from 2006 to Date

#### 2.6. Approximation Algorithms

#### 2.7. When Depots Are Unknown or Unnecessary

- Closed paths (CP).
- Open paths (OP).
- Minsum objective function.
- Minmax objective function.
- Bounding constraints:
- -
- Lower bound on the number of vertices per path.
- -
- Upper bound on the number of vertices per path.

## 3. Integer Programs for DF$\mathit{m}$TSP

#### 3.1. Based on FD-MmTSP

**Definition**

**1.**

**Observation**

**1.**

**Proof.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

- If ${\sum}_{{v}_{k}\in D}{x}_{k,i,k}={\sum}_{{v}_{k}\in D}{x}_{i,k,k}=0$, then $2\le {t}_{i}\le U-1$.
- This case corresponds to the vertices non-adjacent to any dummy depot in any path. In Figure 3, these are ${v}_{3}$, ${v}_{4}$, ${v}_{7}$, and ${v}_{8}$.

- If ${\sum}_{{v}_{k}\in D}{x}_{k,i,k}=1$ and ${\sum}_{{v}_{k}\in D}{x}_{i,k,k}=0$, then ${t}_{i}=1$.
- This case corresponds to the first vertex visited by each salesperson after leaving its dummy depot. In Figure 3, these are ${v}_{1}$ and ${v}_{3}$.

- If ${\sum}_{{v}_{k}\in D}{x}_{k,i,k}=0$ and ${\sum}_{{v}_{k}\in D}{x}_{i,k,k}=1$, then $L\le {t}_{i}\le U$.
- This case corresponds to the last vertex visited by each salesperson before returning to its dummy depot. In Figure 3, these are ${v}_{2}$ and ${v}_{6}$.

**Proof.**

**Proof.**

#### 3.2. More Compact Programs for DFmTSP

**Proof.**

## 4. Empirical Tests

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CP | Closed paths |

CP-DFmTSP | Closed-paths depot-free multiple traveling salesperson problem |

CP-FD-MmTSP | Closed-paths fixed-destination multiple-depots multiple traveling salesperson problem |

DF | Depot free |

DFmTSP | Depot-free multiple traveling salesperson problem |

FD-MmTSP | Fixed-destination multiple-depots multiple traveling salesperson problem |

ILP | Integer linear program |

IP | Integer program |

IQP | Integer quadratic program |

M | Multiple depots |

MmTSP | Multiple-depots multiple traveling salesperson problem |

mTSP | Multiple traveling salesperson problem |

NFD-MmTSP | Non-fixed-destination multiple-depots multiple traveling salesperson problem |

OP | Open paths |

OP-DFmTSP | Open-paths depot-free multiple traveling salesperson problem |

S | Single-depot |

SECs | Subtour elimination constraints |

SmTSP | Single-depot multiple traveling salesperson problem |

TSP | Traveling salesperson problem |

## Appendix A

**Figure A1.**Convergence time reported by Gurobi for the minsum CP-DFmTSP with $L=2$ and tight bounding constraints.

**Figure A2.**Convergence time reported by Gurobi for the minsum CP-DFmTSP with $L=2$ and loose bounding constraints.

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**Figure 1.**Optimal solutions for (

**a**) closed-paths depot-free mTSP (CP-DFmTSP), (

**b**) closed-paths single-depot mTSP (CP-SmTSP), and (

**c**) closed-paths fixed-destination multiple-depots mTSP (CP-FD-MmTSP). The objective function is minsum, the number of salespersons is two ($m=2$), each path must have between three and five vertices (bounding constraints), the cost of each edge equals the euclidean distance between its vertices, and the depots are marked in green. Subfigures (

**d**–

**f**) correspond to the respective open-paths (OP) variants.

**Figure 3.**Exact solution of the IP for minsum CP-DFmTSP with bounding constraints. In this graph instance, $V=\{{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6},{v}_{7},{v}_{8}\}$, $m=2$, $L=3$, $U=5$, $D=\{{v}_{9},{v}_{10}\}$, and the cost of each edge equals the euclidean distance between its vertices (except edges with some dummy depot). There is a $10\times 10$ matrix for each dummy depot, ${\mathbf{x}}_{9}={\left[{x}_{i,j}\right]}_{10\times 10}$ and ${\mathbf{x}}_{10}={\left[{x}_{i,j}\right]}_{10\times 10}$.

**Figure 4.**Optimal solutions for (

**a**) CP-DFmTSP, (

**b**) a combination between CP-DFmTSP and CP-FD-MmTSP ($R=\{{v}_{9},{v}_{13}\}$), and (

**c**) CP-FD-MmTSP ($R=\{{v}_{9},{v}_{13},{v}_{4}\}$). The objective function is minsum, the number of salespersons is three ($m=3$), $L=4$, $U=10$, the cost of each edge equals the euclidean distance between its vertices, and the depots are marked in green. Subfigures (

**d**–

**f**) correspond to the open-paths (OP) variants.

**Figure 5.**Exact solution of the IP for minsum CP-DFmTSP with bounding constraints. In this graph instance, $V=\{{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6},{v}_{7},{v}_{8}\}$, $m=2$, $L=3$, $U=5$, $D=\{{v}_{9},{v}_{10}\}$, and the cost of each edge equals the euclidean distance between its vertices (except edges with some dummy depot). There is only one matrix, $\mathbf{x}={\left[{x}_{i,j}\right]}_{10\times 10}$.

**Figure 6.**Exact solution of the IP for minsum CP-FD-MmTSP with bounding constraints (Section 3.2). In this graph instance, $V=\{{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{6},{v}_{7},{v}_{8}\}$, $m=2$, $L=3$, $U=5$, $R=\{{v}_{2},{v}_{4}\}$, $D=\{{v}_{9},{v}_{10}\}$, and the cost of each edge equals the euclidean distance between its vertices (except edges with some dummy depot). There is only one matrix, $\mathbf{x}={\left[{x}_{i,j}\right]}_{10\times 10}$.

**Figure 7.**Convergence time reported by Gurobi for the instance gr48 for the minsum CP-DFmTSP with $L=2$. Subfigures (

**a**,

**b**) correspond to tight bounding constraints, i.e., $U=\lceil n/m\rceil $. Subfigures (

**c**,

**d**) correspond to loose bounding constraints, i.e., $U=n$.

**Table 1.**Main categories and scope of related work. S, M, and DF stand for single depot, multiple depot, and depot free, respectively.

Main Scope | mTSP | Reference |
---|---|---|

Integer programming | S M DF | [4,5,6,10,11,12,13,14,15,16,17,18,19,20,21,22,49] [6,15,18,42,44,47,48,50,51,52,53,54] [36,51,104] |

Exact algorithm | S M DF | [14,20,23,45,46] [42,43,44,45,47,115] - |

Heuristic | S M DF | [19,24,25,26,27,40,41] [38,39,40] [37,40] |

Metaheuristic | S | [28,29,30,31,34,35,55,56,57,58,59,60,61,62,63,64,65,67,68,69,70,71,72,73,74,76,78,80,81] |

[83,84,85,87,88,90,91,94,95,96,97,98,100,101,102,103,104,105,107,117] | ||

M DF | [82,86,88,89,92,93,99,104,105,118] [32,33,66,75,77,79,104,106] | |

Approximation algorithm | S M DF | [108,116] [108,110,111,112,113,114,115] [109] |

**Table 2.**IPs’ comparison for minsum CP-DFmTSP with $L=2$ and tight bounding constraints. The best-found solutions are bold.

Instance | n | m | U | Karabulut et al. [104] | IQP1 | ILP1 | IQP2 | ILP2 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | ||||

dantzig42 | 42 | 3 | 14 | 748 | 320 | 18% | 772 | 2914 | 24% | 739 | 3934 | 21% | 772 | 1125 | 24% | 739 | 6823 | 21% |

5 | 9 | 754 | 5559 | 21% | 787 | 1843 | 34% | 786 | 2639 | 34% | 879 | 4030 | 43% | 1055 | 1763 | 52% | ||

swiss42 | 42 | 3 | 14 | 1410 | 5682 | 17% | 1407 | 1155 | 21% | 1629 | 6377 | 33% | 1430 | 1043 | 24% | 1616 | 107 | 33% |

5 | 9 | 1397 | 886 | 19% | 1603 | 4673 | 38% | 1394 | 1741 | 26% | 1470 | 1493 | 37% | 1787 | 246 | 48% | ||

att48 | 48 | 3 | 16 | 12,557 | 7200 | 25% | 10,941 | 859 | 17% | 12,339 | 777 | 26% | 12,683 | 6329 | 31% | 12,432 | 668 | 28% |

5 | 10 | - | - | - | 11,149 | 2272 | 27% | 11,505 | 1884 | 28% | 11,748 | 4602 | 33% | 16,160 | 7129 | 52% | ||

gr48 | 48 | 3 | 16 | 5337 | 4452 | 13% | 5213 | 2881 | 15% | 5478 | 528 | 21% | 5423 | 1724 | 23% | 6030 | 5921 | 30% |

5 | 10 | - | - | - | 7009 | 6895 | 45% | 5530 | 7184 | 30% | 6696 | 2286 | 44% | 7060 | 4639 | 47% | ||

hk48 | 48 | 3 | 16 | 11,999 | 1656 | 9% | 12,620 | 663 | 20% | 12,568 | 6956 | 19% | 13,576 | 643 | 26% | 12,456 | 4357 | 20% |

5 | 10 | - | - | - | 13,546 | 5583 | 33% | - | - | - | - | - | - | 15,316 | 6864 | 42% |

**Table 3.**IPs’ comparison for minsum CP-DFmTSP with $L=2$ and loose bounding constraints. The best-found solutions are bold.

Instance | n | m | U | Karabulut et al. [104] | IQP1 | ILP1 | IQP2 | ILP2 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | $\mathit{f}$ | $\mathit{t}\mathbf{\left(}\mathit{s}\mathbf{\right)}$ | $\mathit{g}$ | ||||

dantzig42 | 42 | 3 | 42 | 633 | 74 | 0% | 633 | 206 | 5.69% | 633 | 91 | 7.11% | 633 | 100 | 3.95% | 633 | 40 | 3.79% |

5 | 42 | 604 | 84 | 0% | 604 | 36 | 4.64% | 604 | 38 | 14.07% | 604 | 135 | 15.07% | 605 | 83 | 13.22% | ||

swiss42 | 42 | 3 | 42 | 1208 | 188 | 0% | 1208 | 89 | 2.15% | 1208 | 27 | 1.57% | 1208 | 203 | 9.11% | 1208 | 27 | 9.27% |

5 | 42 | 1155 | 50 | 1.47% | 1155 | 413 | 5.11% | 1155 | 812 | 5.63% | 1167 | 2200 | 19.37% | 1155 | 60 | 17.49% | ||

att48 | 48 | 3 | 48 | 9946 | 442 | 4.25% | 9946 | 11 | 5.81% | 9946 | 85 | 4.83% | 9946 | 51 | 7.57% | 9946 | 26 | 7.51% |

5 | 48 | 9448 | 43 | 2.64% | 9448 | 182 | 13.52% | 9448 | 400 | 13.79% | 9448 | 85 | 15.81% | 9448 | 2000 | 14.73% | ||

gr48 | 48 | 3 | 48 | 4761 | 201 | 1.70% | 4761 | 96 | 3.91% | 4761 | 27 | 6.87% | 4761 | 1369 | 8.36% | 4761 | 227 | 8.25% |

5 | 48 | 4544 | 113 | 0% | 4544 | 207 | 8.78% | 4558 | 71 | 7.79% | 4735 | 171 | 18.59% | 4558 | 2829 | 14.26% | ||

hk48 | 48 | 3 | 48 | 11,101 | 115 | 0% | 11,101 | 335 | 2.08% | 11,101 | 206 | 2.20% | 11,332 | 3037 | 10.98% | 11,134 | 1950 | 8.45% |

5 | 48 | 10,834 | 164 | 0% | 10,834 | 919 | 6.98% | 10,834 | 786 | 7.30% | 10,888 | 1524 | 17.20% | 10,967 | 127 | 17.42% |

IP | Section | Objective Function | Binary Variables | Constraints |
---|---|---|---|---|

IQP1 | Section 3.1 | quadratic | $\mathcal{O}\left({n}^{2}m\right)$ | $\mathcal{O}\left({n}^{2}\right)$ |

ILP1 | linear | $\mathcal{O}\left({n}^{2}m\right)$ | $\mathcal{O}\left({n}^{2}m\right)$ | |

IQP2 | Section 3.2 | quadratic | $\mathcal{O}\left({n}^{2}\right)$ | $\mathcal{O}\left({n}^{2}\right)$ |

ILP2 | linear | $\mathcal{O}\left({n}^{2}\right)$ | $\mathcal{O}\left({n}^{2}m\right)$ |

IP | Dummy Depots | CP | OP | Minsum | Minmax | L | U | FD-M+DF |
---|---|---|---|---|---|---|---|---|

IQP1 | m | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | |

ILP1 | m | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 |

IQP2 | m | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | ||

ILP2 | m | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 | 🗸 |

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**MDPI and ACS Style**

Cornejo-Acosta, J.A.; García-Díaz, J.; Pérez-Sansalvador, J.C.; Segura, C.
Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems. *Mathematics* **2023**, *11*, 3014.
https://doi.org/10.3390/math11133014

**AMA Style**

Cornejo-Acosta JA, García-Díaz J, Pérez-Sansalvador JC, Segura C.
Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems. *Mathematics*. 2023; 11(13):3014.
https://doi.org/10.3390/math11133014

**Chicago/Turabian Style**

Cornejo-Acosta, José Alejandro, Jesús García-Díaz, Julio César Pérez-Sansalvador, and Carlos Segura.
2023. "Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems" *Mathematics* 11, no. 13: 3014.
https://doi.org/10.3390/math11133014