We consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a \((2 + \frac{2}{7})\)-approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a \(2 + \epsilon \) lower bound for the relaxation.

我们考虑有能力的循环覆盖问题：给定一个无向、完全图G，具有度量边长和对顶点的需求，我们希望用顶点不相交循环覆盖顶点，每个循环最多满足一个需求。目标是最小化总长度和循环数的线性组合。这个问题与有能力的车辆路径问题 (CVRP) 和其他循环覆盖问题密切相关，例如最小-最大循环覆盖和有界循环覆盖。我们表明，通过将解决方案与多拟阵松弛进行比较，贪婪算法和后处理步骤产生了\((2 + \frac{2}{7})\) - 近似值。我们还表明对我们算法的分析是严密的，并提供了一个\(2 + \epsilon \)松弛下限。