A cycle cover of a graph is a spanning subgraph whose connected components are simple cycles. Given a complete weighted directed graph, consider the intractable problem of finding a maximum-weight cycle cover which satisfies an upper bound on the number of cycles and a lower bound on the number of edges in each cycle. We suggest a polynomial-time algorithm for solving this problem in the geometric case where the vertices of the graph are points in a multidimensional real space and the distances between them are induced by a positively homogeneous function whose unit ball is an arbitrary convex polytope with a fixed number of facets. The obtained result extends the ideas underlying the well-known algorithm for the polyhedral Max TSP.

中文翻译：

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约束循环数和长度的多面循环覆盖问题算法

图的循环覆盖是一个跨接子图，其连接的组件为简单循环。给定一个完整的加权有向图，请考虑一个难于解决的问题，即找到一个最大权重的循环覆盖数，该覆盖数满足每个循环的循环数的上限和下限的边缘数。我们建议在几何情况下解决该问题的多项式时间算法，在这种情况下，图形的顶点是多维实空间中的点，并且它们之间的距离是由一个正齐次函数引起的，该函数的单位球是一个任意凸多面体，具有固定数量的构面。获得的结果扩展了多面体Max TSP的著名算法基础的思想。