In this paper we study a problem of determining when entities are active based on their interactions with each other. We consider a set of entities V and a sequence of time-stamped edges E among the entities. Each edge \((u,v,t)\in E\) denotes an interaction between entities u and v at time t. We assume an activity model where each entity is active during at most k time intervals. An interaction (u, v, t) can be explained if at least one of u or v are active at time t. Our goal is to reconstruct the activity intervals for all entities in the network, so as to explain the observed interactions. This problem, the network-untangling problem, can be applied to discover event timelines from complex entity interactions. We provide two formulations of the network-untangling problem: (i) minimizing the total interval length over all entities (sum version), and (ii) minimizing the maximum interval length (max version). We study separately the two problems for \(k=1\) and \(k>1\) activity intervals per entity. For the case \(k=1\), we show that the sum problem is NP-hard, while the max problem can be solved optimally in linear time. For the sum problem we provide efficient algorithms motivated by realistic assumptions. For the case of \(k>1\), we show that both formulations are inapproximable. However, we propose efficient algorithms based on alternative optimization. We complement our study with an evaluation on synthetic and real-world datasets, which demonstrates the validity of our concepts and the good performance of our algorithms.

在本文中，我们研究了根据实体之间的相互作用来确定实体何时处于活动状态的问题。我们认为一个实体的集合V和时间冲压边沿的序列Ë的实体之间。E（）中的每个边\（（u，v，t）\）表示实体u和v在时间t处的交互。我们假设一个活动模型，其中每个实体最多在k个时间间隔内处于活动状态。如果u或v中至少一个在时间t处于活动状态，则可以解释交互作用（u，  v，  t）。我们的目标是重建网络中所有实体的活动间隔，以解释观察到的交互。这个问题，即网络纠缠问题，可以应用于从复杂的实体交互中发现事件时间表。我们提供了两种解决网络纠缠问题的方法：（i）最小化所有实体的总间隔长度（求和版本），以及（ii）最小化最大间隔长度（最大版本）。我们分别研究每个实体\（k = 1 \）和\（k> 1 \）活动间隔的两个问题。对于\（k = 1 \）的情况，我们证明和问题是NP-hard，而最大问题可以在线性时间内最佳解决。对于总和问题，我们提供了以实际假设为动力的有效算法。对于\（k> 1 \）的情况，我们表明两种公式都是不可近似的。但是，我们提出了基于替代优化的高效算法。我们通过对合成数据集和真实数据集进行评估来补充我们的研究，这证明了我们的概念的有效性和算法的良好性能。