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Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists^{ †}

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Worst case example for the Gale–Shapley algorithm for $n=5$. Partners in the man-optimal stable matching are underlined. Each preference list is ordered from left to right in increasing order of the rank, i.e., the leftmost person is the most preferable and the rightmost person is the least preferable.

## 2. Preliminaries

#### 2.1. Problem Definitions

#### 2.2. Reduced Lists and Rotation Digraphs

## 3. Changing One Man’s Preference List

#### 3.1. Optimization Variant

**Proposition 3.1**

**Example.**

Algorithm 1: |

1: Find the man-optimal stable matching M_{0} of I. |

2: for each man m do |

3: Modify I by moving M_{0}(m) to the top of m’s list. Let I_{m} be the resulting instance. |

4: Apply GS to I_{m} and find its man-optimal stable matching M_{m}. |

5: end for |

6: Let m* be a man such that the score of M_{m*} is minimum. |

7: Output I_{m*} . |

**for**loop can be executed in time $O\left({n}^{2}\right)$. Since there are n men, the overall time complexity of Algorithm 1 is $O\left({n}^{3}\right)$.

**Theorem 3.2**

#### 3.2. Decision Variant

Algorithm 2: |

1: Find the man-optimal stable matching M_{0} of I. |

2: Construct D_{I,}_{M0} . |

3: for each man m do |

4: Modify I by moving M_{0}(m) to the top of m’s list. Let I_{m} be the resulting instance. |

5: Construct D_{Im,}_{M0} . |

6: if D_{Im,}_{M0} contains a cycle then |

7: Output “yes” and terminate. |

8: end if |

9: end for |

10: Output “no”. |

**Example.**We give an example of the execution of Algorithm 2 for instance I of Figure 1. The man-optimal stable matching ${M}_{0}$ is depicted in Figure 1. In constructing the reduced lists $R{L}_{I,{M}_{0}}$, we delete ${w}_{5}$ from ${m}_{2},{m}_{3},{m}_{4}$, and ${m}_{5}$’s lists, and ${m}_{2},{m}_{3},{m}_{4}$, and ${m}_{5}$ from ${w}_{5}$’s list. Hence $R{L}_{I,{M}_{0}}$ is as follows:

m_{1}: w_{1} w_{2} w_{3} w_{4} w_{5} | w_{1}: m_{2} m_{3} m_{4} m_{5} m_{1} |

m_{2}: w_{2} w_{3} w_{4} w_{1} | w_{2}: m_{3} m_{4} m_{5} m_{1} m_{2} |

m_{3}: w_{3} w_{4} w_{1} w_{2} | w_{3}: m_{4} m_{5} m_{1} m_{2} m_{3} |

m_{4}: w_{4} w_{1} w_{2} w_{3} | w_{4}: m_{5} m_{1} m_{2} m_{3} m_{4} |

m_{5}: w_{1} w_{2} w_{3} w_{4} | w_{5}: m_{1} |

m_{1}: w_{1} w_{2} w_{3} w_{4} w_{5} | w_{1}: m_{2} m_{3} m_{4} m_{1} |

m_{2}: w_{2} w_{3} w_{4} w_{1} | w_{2}: m_{3} m_{4} m_{1} m_{2} |

m_{3}: w_{3} w_{4} w_{1} w_{2} | w_{3}: m_{4} m_{1} m_{2} m_{3} |

m_{4}: w_{4} w_{1} w_{2} w_{3} | w_{4}: m_{5} m_{1} m_{2} m_{3} m_{4} |

m_{5}: w_{4} | w_{5}: m_{1} |

m_{1}: w_{5} | w_{1}: m_{2} m_{3} m_{4} m_{5} |

m_{2}: w_{2} w_{3} w_{4} w_{1} | w_{2}: m_{3} m_{4} m_{5} m_{2} |

m_{3}: w_{3} w_{4} w_{1} w_{2} | w_{3}: m_{4} m_{5} m_{2} m_{3} |

m_{4}: w_{4} w_{1} w_{2} w_{3} | w_{4}: m_{5} m_{2} m_{3} m_{4} |

m_{5}: w_{1} w_{2} w_{3}w_{4} | w_{5}: m_{1} |

**Theorem 3.3**

## 4. Changing k Men’s Preference Lists

#### 4.1. Hardness Result

**Theorem 4.1**

m_{i}: L(W_{i}) w_{i} … | (1 ≤ i ≤ |V|) | w_{i}: m_{i} … | (1 ≤ i ≤ |V|) |

m_{i,j}: ${w}_{i,j}^{\prime}$ w_{i,j} … | ((i, j) ∈ E) | w_{i,j}: m_{i,j} m_{i} m_{j} ${m}_{i,j}^{\prime}$ … | ((i, j) ∈ E) |

${m}_{i,j}^{\prime}$: w_{i,j} ${w}_{i,j}^{\prime}$ … | ((i, j) ∈ E) | ${w}_{i,j}^{\prime}$: ${m}_{i,j}^{\prime}$ m_{i, j} … | ((i,j) ∈ E) |

#### 4.2. Optimization Variant

**Figure 3.**A counter example for the naive algorithm where $k=2$. Man-optimal partners are underlined. An optimal solution is obtained by moving ${w}_{4}$ and ${w}_{5}$ to the top in ${m}_{4}$’s and ${m}_{5}$’s preference lists respectively, as a result of which the score decreases by seven. Any choice of two men and moving their man-optimal partners to the top decreases the score by at most five.

**Lemma 4.2**

**Theorem 4.3**

Algorithm 3: |

1: Find the man-optimal stable matching M_{0} of I. |

2: for each man m do |

3: Moving M_{0}(m) to the top of m’s list. |

4: for each choice of k ‒ 1 men (m_{1}, … , m_{k‒1}) from X ‒ {m} do |

5: for each combination of k ‒ 1 women (w_{1}, … , w_{k‒1}) such that each w_{i} is M_{0}(m_{i}) or precedes M_{0}(m_{i}) in m_{i}’s list of I do |

6: Move w_{i} to the top of m_{i}’s preference list. |

7: end for |

8: Apply GS to the current instance and find its man-optimal stable matching. |

9: end for |

10: end for |

11: Output the instance that minimizes the man-optimal score. |

#### 4.3. Decision Variant

**Theorem 4.4**

## 5. Changing n Men’s Preference Lists

**Theorem 5.1**

**Theorem 5.2**

## 6. Conclusions

## Acknowledgements

## References

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## Share and Cite

**MDPI and ACS Style**

Inoshita, T.; Irving, R.W.; Iwama, K.; Miyazaki, S.; Nagase, T.
Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists. *Algorithms* **2013**, *6*, 371-382.
https://doi.org/10.3390/a6020371

**AMA Style**

Inoshita T, Irving RW, Iwama K, Miyazaki S, Nagase T.
Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists. *Algorithms*. 2013; 6(2):371-382.
https://doi.org/10.3390/a6020371

**Chicago/Turabian Style**

Inoshita, Takao, Robert W. Irving, Kazuo Iwama, Shuichi Miyazaki, and Takashi Nagase.
2013. "Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists" *Algorithms* 6, no. 2: 371-382.
https://doi.org/10.3390/a6020371