We study the problem of partitioning a weighted graph into connected components such that each component fulfills upper and lower weight constraints. Partitioning into a minimum, maximum or a fixed number of clusters is NP-hard in general but polynomial-time solvable on trees. In this paper, we present a polynomial-time algorithm for cactus graphs. For other optimization goals or additional constraints, the partition problem becomes NP-hard even on trees and for a lower weight bound equal to zero. We show that our method can be used as an algorithmic framework to solve other partition problems for cactus graphs in pseudo-polynomial time.

中文翻译：

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加权仙人掌图的分区算法

我们研究将加权图划分为连接组件的问题，以便每个组件满足上下权重约束。划分为最小、最大或固定数量的簇通常是 NP-hard 问题，但在树上可在多项式时间内求解。在本文中，我们提出了仙人掌图的多项式时间算法。对于其他优化目标或附加约束，分区问题即使在树上也变得 NP-hard 并且权重边界为零。我们表明，我们的方法可以用作算法框架来解决伪多项式时间内仙人掌图的其他分区问题。