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：对偶线性规划理论从早期到现代，一直对博弈核心的研究起着核心作用。然而，尽管这方面的工作广泛而深入，但仍存在基本空白。我们利用对偶线性规划理论中的以下构建模块来填补这些空白：1. 全幺模性（TUM）。2. 互补松弛条件和严格互补性。对全幺模性的探索使我们能够定义新的博弈、刻画其核心，并创新性地利用核心分配来强制执行这些博弈应用中自然产生的约束。后者包括：1. 寻找最小最大公平、最大最小公平和公平核心分配的高效算法。2. 在指派博弈的推广中鼓励多样性并避免过度代表。互补性使我们能够证明指派博弈及其推广中核心分配的新性质。