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 and max-min fair 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一直在博弈核心（core）的研究中扮演着核心角色。然而，尽管这一领域的研究工作广泛深入，基本空白仍然存在。我们利用LP-对偶理论中的以下构建模块填补这些空白：1. 全幺模性（Total Unimodularity，TUM）；2. 互补松弛条件与严格互补性。对全幺模性的探索使我们定义新博弈、表征其核心，并开创性地运用核心分配（core imputations）来强制执行这些博弈应用中自然出现的约束。后者包括：1. 寻找极小极大公平与极大极小公平核心分配的高效算法；2. 在指派博弈（assignment game）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指派博弈及其推广形式的核心分配的新性质。