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 [KL85] 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 [KL85]. Then, we show our generalization is defined by a Galois connection between the greedoid flats and closed sets of a representation. Finally, we apply this duality to identify a subclass of polymatroid greedoids with favorable properties, which we call strong polymatroid greedoids. 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：摘要：贪心拟阵是拟阵的一种推广形式，能够对组合优化问题进行更灵活的分析和建模。然而，这类结构削弱了许多使拟阵具有普适性的核心性质。多拟阵贪心拟阵[KL85]在此之间提供了一个有趣的中间地带，因此我们进一步拓展了该类别。首先，我们证明每个局部偏序集贪心拟阵（若其贪心算法能正确求解基本词上的线性优化问题）必然具有多拟阵表示。为此，我们利用贪心拟阵闭包格与多拟阵闭集格之间的关系，推广了[KL85]中的概念。随后，我们证明该推广由贪心拟阵闭包与表示闭集之间的伽罗瓦连接所定义。最后，我们运用这一对偶性，识别出多拟阵贪心拟阵中具有优良性质的一个子类，称之为强多拟阵贪心拟阵。作为分析的技术工具，我们引入了区间贪心拟阵的乐观性与分叉引理。这两个工具贯穿全文且具有独立的研究价值。