A greedoid is a generalization of a matroid allowing for more flexible analyses and modeling of combinatorial optimization problems. However, these structures decimate many matroid properties contributing to their pervasive nature. A polymatroid greedoid [KL85a] presents an interesting middle ground, so we further develop this class. First we prove every local poset greedoid for which the greedy algorithm correctly solves linear optimizations over its basic words must have a polymatroid representation. For this, we use relationships between the lattices of greedoid flats and closed sets of a polymatroid to generalize concepts in [KL85a]. Then, we show our generalization induces a Galois injection between the greedoid flats and closed sets of a representation. Finally, we apply this duality to identify a subclass of polymatroid greedoids with a maximum representation, giving a partial answer to an open problem of [KL85a]. As technical tools for our analyses, we introduce optimism and the Forking Lemma for interval greedoids. Both are pervasive in our work, and are of independent interest.

翻译：贪婪拟阵是拟阵的推广，允许对组合优化问题进行更灵活的分析与建模。然而，这些结构削弱了许多构成拟阵普遍性质的特征。多拟阵贪婪拟阵[KL85a]提供了一个有趣的中介类别，因此我们进一步拓展此类结构。首先，我们证明：对于任一局部偏序集贪婪拟阵，若贪心算法能正确求解其基本词上的线性优化问题，则该贪婪拟阵必具有多拟阵表示。为此，我们利用贪婪拟阵平坦格与多拟阵闭集格之间的关系，推广了[KL85a]中的概念。接着，我们证明该推广在贪婪拟阵平坦与表示的闭集之间诱导出一个伽罗瓦单射。最后，我们应用此对偶性识别出一类具有最大表示的多拟阵贪婪拟阵子类，从而部分回答了[KL85a]中的一个开放问题。作为分析的技术工具，我们引入了区间贪婪拟阵的乐观性引理与分叉引理。二者在我们的工作中贯穿始终，且具有独立的研究价值。