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 algorithm (TBLS) 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-sized instances.

本研究针对稳定婚姻问题及医院/居民问题中的不完全列表和决胜情况，提出了一种基于决胜的局部搜索算法（TBLS）。该算法通过调节偏好顺序，在最大化匹配规模的同时，推出了一种关注公平性的变体TBLS-E，显著提升了性别平等性并保持了较高的匹配规模。与现有算法相比，TBLS和TBLS-E在计算速度和匹配效果上均表现出色。

基于决胜的局部搜索算法解决稳定匹配问题