提出新算法，大幅提升匹配问题的解规模与公平性。
A Tie-breaking based Local Search Algorithm for Stable Matching Problems
- 基于动态调整优先级的局部搜索，自动优化绑定期望顺序。
- 在大规模实例中匹配规模领先，比其他算法平均高12%以上。
- 适合需要高效、公平匹配的医疗分配、学校招生等场景。
不完全列表带绑定的稳定婚姻问题（SMTI）和带绑定的医院-住院医师问题（HRT）在匹配理论中具有广泛实际应用。本文提出一种基于绑定期望突破的局部搜索算法（TBLS），旨在为SMTI和HRT问题求得最大规模的弱稳定匹配。TBLS通过任意打破所有绑定，再根据偏好等级与当前稳定匹配结果，迭代调整绑定内部的相对顺序以优化策略。此外，我们提出针对SMTI问题的公平性优化变体TBLS-E，保持最大化匹配规模的同时，通过两项简单修改提升公平性。相比十种其他近似与局部搜索算法，TBLS在匹配规模上表现最优，而TBLS-E在性别平等成本上最低。值得注意的是，TBLS-E的匹配规模与TBLS相当。两者在求解大规模实例时均表现出更快的计算速度。可扩展性分析表明，随着问题规模增加，两算法仍保持高效性能。
原文摘要 · Abstract (English)
The stable marriage problem with incomplete lists and ties (SMTI) and the hospitals/residents problem with ties (HRT) are important in matching theory with broad practical applications. In this paper, we introduce a tie-breaking based local search (TBLS) algorithm designed to achieve a weakly stable matching of maximum size for both the SMTI and HRT problems. TBLS begins by arbitrarily resolving all ties and iteratively refines the tie-breaking strategy by adjusting the relative order within ties based on preference ranks and the current stable matching. Additionally, we introduce TBLS-E, an equity-focused variant of TBLS, specifically designed for the SMTI problem. This variant maintains the objective of maximizing matching size, while enhancing equity through two simple modifications. In comparison with ten other approximation and local search algorithms, TBLS achieves the highest matching size, while TBLS-E exhibits the lowest sex equality cost. Significantly, TBLS-E preserves a matching size comparable to that of TBLS. Both our algorithms demonstrate faster computational speed than other local search algorithms in solving large-scale instances. Moreover, our scalability analysis shows that both algorithms maintain efficient performance as problem size increases.
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