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: \begin{enumerate} \item Total unimodularity (TUM). \item Complementary slackness conditions and strict complementarity. \end{enumerate} 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: \begin{enumerate} \item Efficient algorithms for finding {\em min-max fair and max-min fair core imputations}. \item {\em Encouraging diversity and avoiding over-representation} in a generalization of the assignment game. \end{enumerate} 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核心的研究中扮演核心角色。然而，尽管这项工作已广泛开展，但基本空白仍然存在。我们利用LP-对偶理论中的以下构建模块来填补这些空白：\begin{enumerate} \item 全幺模性（Total Unimodularity, TUM）。\item 互补松弛条件与严格互补性。\end{enumerate} 对TUM的探索促使我们定义新的博弈，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出现的约束。后者包括：\begin{enumerate} \item 寻找{\em 极小极大公平和极大极小公平核心分配}的高效算法。\item 在分配博弈的推广中{\em 促进多样性并避免过度代表性}。\end{enumerate} 互补性使我们能够证明分配博弈及其推广的核心分配的新性质。