The Stable Marriage problem (SM), solved by the famous deferred acceptance algorithm of Gale and Shapley (GS), has many natural generalizations. If we allow ties in preferences, then the problem of finding a maximum solution becomes NP-hard, and the best known approximation ratio is $1.5$ (McDermid ICALP 2009, Paluch WAOA 2011, Z. Kir\'aly MATCH-UP 2012), achievable by running GS on a cleverly constructed modified instance. Another elegant generalization of SM is the matroid kernel problem introduced by Fleiner (IPCO 2001), which is solvable in polynomial time using an abstract matroidal version of GS. Our main result is a simple $1.5$-approximation algorithm for the matroid kernel problem with ties. We also show that the algorithm works for several other versions of stability defined for cardinal preferences, by appropriately modifying the instance on which GS is executed. The latter results are new even for the stable marriage setting.

翻译：稳定婚姻问题（SM）由Gale和Shapley（GS）提出的著名延迟接受算法求解，其存在许多自然推广。若允许偏好中存在平局，则寻找最大解的问题变为NP难问题，目前已知的最佳近似比为1.5（McDermid ICALP 2009，Paluch WAOA 2011，Z. Király MATCH-UP 2012），该结果可通过在巧妙构造的修改实例上运行GS算法实现。SM的另一个优雅推广是Fleiner提出的拟阵核问题（IPCO 2001），通过运用抽象的拟阵化GS算法可在多项式时间内求解。本文的核心成果是为带平局的拟阵核问题设计了一个简单的1.5-近似算法。我们同时证明，通过适当修改执行GS算法的实例，该算法可适用于多种基于基数偏好定义的稳定性变体。即使对于稳定婚姻场景，后者结果亦属全新。