A class of counting problems asks for the number of regions of a central hyperplane arrangement. By duality, this is the same as counting the vertices of a zonotope. Efficient algorithms are known that solve this problem by computing the vertices of a zonotope from its set of generators. Here, we give an efficient algorithm, based on a linear optimization oracle, that performs the inverse task and recovers the generators of a zonotope from its set of vertices. We also provide a variation of that algorithm that allows to decide whether a polytope, given as its vertex set, is a zonotope and when it is not a zonotope, to compute its greatest zonotopal summand.

一类计数问题要求计算中心超平面排列的区域数。根据对偶性，这与计算带位环的顶点相同。已知有效的算法通过从一组生成器计算带位的顶点来解决这个问题。在这里，我们给出了一种基于线性优化预言机的高效算法，该算法执行逆向任务并从其顶点集恢复带状环的生成器。我们还提供了该算法的一个变体，它允许决定作为其顶点集给出的多面体是否是zonotope，以及当它不是zonotope时，计算其最大的zonotopal summand。