Optimizing an implicational base of a closure system consists in turning this implicational base into an equivalent one with premises and conclusions as small as possible. This task is known to be hard in general but tractable for a number of classes of closure systems. In particular, several classes of convex geometries are known to have tractable optimization, while the problem was recently claimed to remain hard in general convex geometries. Continuing this line of research, we give a characterization of the optimum bases of a convex geometry in terms of what we call quasi-closed hypergraphs. We then use this characterization to show that when each quasi-closed hypergraph has disjoint edges, any implicational base of the convex geometry can be optimized in polynomial time with existing minimization and reduction algorithms. Finally, we prove that this property applies to double-shelling, acyclic, affine and acceptant convex geometries, thus unifying the existing results regarding the tractability of optimization for the first three classes.

翻译：优化闭包系统的蕴含基旨在将其转化为等价的、前提与结论尽可能小的基。已知该问题在一般情况下难以处理，但对于若干类闭包系统是可处理的。特别地，已知若干类凸几何具有可处理的优化问题，而近期有研究声称该问题在一般凸几何中仍保持困难。延续这一研究方向，我们通过定义拟闭超图给出了凸几何最优基的特征刻画。利用该特征，我们证明了当每个拟闭超图具有不相交边时，凸几何的任何蕴含基均可通过现有的最小化与约简算法在多项式时间内优化。最后，我们证明该性质适用于双重壳形、无圈、仿射及接受型凸几何，从而统一了前三类凸几何优化可处理性的已有结论。