Monotone trees - trees with a function defined on their vertices that decreases the further away from a root node one travels, are a natural model for a process that weakens the further one gets from its source. Given an aggregation of monotone trees, one may wish to reconstruct the individual monotone components. A natural representation of such an aggregation would be a graph. While many methods have been developed for extracting hidden graph structure from datasets, which makes obtaining such an aggregation possible, decomposing such graphs into the original monotone trees is algorithmically challenging. Recently, a polynomial time algorithm has been developed to extract a minimum cardinality collection of monotone trees (M-Tree Set) from a given density tree - but no such algorithm exists for density graphs that may contain cycles. In this work, we prove that extracting such minimum M-Tree Sets of density graphs is NP-Complete. We additionally prove three additional variations of the problem - such as the minimum M-Tree Set such that the intersection between any two monotone trees is either empty or contractible (SM-Tree Set) - are also NP-Complete. We conclude by providing some approximation algorithms, highlighted by a 3-approximation algorithm for computing the minimum SM-Tree Set for density cactus graphs.

翻译：单调树——一种在其顶点上定义函数且函数值随距离根节点越远而递减的树——是描述过程随远离源头而减弱的自然模型。给定单调树的聚合，人们可能希望重建各个单调分量。这种聚合的自然表示形式是图。尽管已有许多方法用于从数据集中提取隐藏的图结构使获得此类聚合成为可能，但将这些图分解为原始单调树在算法上具有挑战性。近期，已有多项式时间算法从给定密度树中提取最小基数单调树集合（M-树集），但对于可能包含环路的密度图尚不存在此类算法。本研究证明，从密度图中提取此类最小M-树集问题是NP完全的。此外，我们还证明了该问题的三个变体（例如任意两个单调树的交集为空或可缩的最小M-树集，即SM-树集）同样是NP完全的。最后，我们提供若干近似算法，重点给出了计算密度仙人掌图的最小SM-树集的3-近似算法。