LP-duality theory has played a central role in the study of the core, right from its early days to the present time. However, despite the extensive nature of this work, basic gaps still remain. We address these gaps using the following building blocks from LP-duality theory: 1. Total unimodularity (TUM). 2. Complementary slackness conditions and strict complementarity. Our exploration of TUM leads to defining new games, characterizing their cores and giving novel ways of using core imputations to enforce constraints that arise naturally in applications of these games. The latter include: 1. Efficient algorithms for finding min-max fair, max-min fair and equitable core imputations. 2. Encouraging diversity and avoiding over-representation in a generalization of the assignment game. Complementarity enables us to prove new properties of core imputations of the assignment game and its generalizations.

翻译：LP-对偶理论从早期至今在博弈核的研究中始终扮演着核心角色。然而，尽管相关研究已相当广泛，基本空白依然存在。我们利用LP-对偶理论中的以下基础构件来填补这些空白：1. 全单模性（TUM）。2. 互补松弛条件与严格互补性。对TUM的探索引导我们定义了新的博弈，刻画了它们的核，并提出了利用核分配来强制执行这些博弈应用中自然产生的约束的新方法。后者包括：1. 寻找最小-最大公平、最大-最小公平和公平核分配的高效算法。2. 在赋值博弈的推广中鼓励多样性并避免过度代表性。互补性使我们能够证明赋值博弈及其推广形式的核分配的新性质。