[论文解读] Local search for stable marriage problems

Mirco Gelain, Maria Pini+3 more|arXiv (Cornell University)|Jul 6, 2010

Game Theory and Voting Systems参考文献 12被引用 4

一句话总结

本文提出一种用于求解稳定婚姻（SM）和带环及不完整列表的稳定婚姻（SMTI）问题的局部搜索算法。通过利用问题特定约束来减少邻域大小并加速局部移动，该方法能高效找到稳定匹配——在SM问题中实现O(n log n)的时间复杂度，并在SMTI问题中尽管其为NP难问题，仍能生成高基数匹配。

ABSTRACT

The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving a SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these lists, an we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We evaluate empirically our algorithm for SM problems by measuring its runtime behaviour and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behaviour and its ability to find a maximum cardinality stable marriage.For SM problems, the number of steps of our algorithm grows only as O(nlog(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages.Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size despite the NP-hardness of this problem.

研究动机与目标

• 解决在具有偏好列表的双边市场中寻找稳定匹配的计算挑战。
• 处理偏好不完整且允许存在并列关系的NP难SMTI变体，目标是最大化匹配规模。
• 开发一种局部搜索框架，以高效探索解空间，同时遵守稳定性约束。
• 从运行时间可扩展性与采样质量两个方面评估算法在所有稳定匹配中的性能表现。

提出的方法

• 将SM和SMTI问题建模为在稳定匹配上的局部搜索优化任务。
• 基于以男性或女性为导向的交换操作（例如，通过循环轮换配偶）定义邻域结构，以保持稳定性。
• 应用剪枝技术排除不稳定或无效的移动，从而在不损失完备性的前提下减少邻域大小。
• 采用迭代改进策略：在每一步中，选择能改善定义目标（如基数或公平性）的邻近稳定匹配。
• 利用稳定匹配的格结构引导搜索过程，提升采样分布的均匀性。
• 通过受限匹配约束处理并列关系与不完整列表，确保仅考虑有效配对。

实验结果

研究问题

• RQ1 在经典SM问题中，局部搜索能否高效地遍历稳定匹配空间？
• RQ2 随着问题规模增大，该局部搜索方法在运行时间上的可扩展性如何？
• RQ3 该算法在SM实例中能多大程度上对稳定匹配的完整格结构进行采样？
• RQ4 在NP难的SMTI问题中，局部搜索能否有效找到高基数的稳定匹配？
• RQ5 尽管问题具有计算困难性，该算法在大规模SMTI实例上的表现如何？

主要发现

• 对于稳定婚姻（SM）问题，该局部搜索算法的步骤数呈现O(n log n)的增长趋势，表明其具有出色的可扩展性。
• 该算法对所有稳定婚姻的格结构采样非常有效，展现出解分布的公平性与均匀性。
• 在SMTI问题中，该算法能快速返回规模大且通常为最优的稳定匹配，即使在大规模实例上亦然。
• 尽管SMTI问题为NP难，该局部搜索方法仍能高效处理大规模问题，在实践中优于传统精确方法。
• 通过使用问题特定的剪枝技术，显著减少了邻域大小，从而在不损失解质量的前提下实现了更快的收敛速度。