## 2 Problem Formulation  
Let \(G=(V,E)\) be a temporal graph, where \(V\) represents the set of nodes and \(E\) denotes the temporal edges. Each edge is characterized by a triple \((u,v,t)\in E\), with \((u,v)\in V\) and \(t\in T\) indicating the timestamp at which the interaction between \(u\) and \(v\) occurs. We focus on undirected and unweighted temporal graphs. For any vertex \(u\in V\), we define \(E(u)=\{(u,v,t)\in E\}\) as the set of temporal edges incident to \(u\). Similarly, \(N(u,t)=\{v\mid(u,v,t)\in E\}\) represents the set of neighbors of \(u\) at time \(t\), while \(T(u)=\{t\mid(u,v,t)\in E\}\) captures the set of timestamps where \(u\) participates in an edge. The degree of a vertex \(u\) at timestamp \(t\) is defined as \(\textit{degree}((u,t))=|N(u,t)|\), which quantifies the number of temporal edges involving \(u\) at time \(t\). We define the _density_ of a temporal graph as the ratio of existing edges to the maximal possible number of edges, expressed as \(d=\frac{2|E|}{|V|(|V|-1)|T|}\). A graph is regarded as sparse if the number of edges satisfies \(|E|=O(|V|+|T|)\). Given two values \(s_{u}\) and \(e_{u}\), where \(s_{u}\leq e_{u}\), we define the activity interval of vertex \(u\) as \(I_{u}=[s_{u},e_{u}]\). The set of such intervals for all vertices, denoted as \(\mathcal{T}=\{I_{u}\}_{u\in V}\), forms an activity timeline of \(G\). The _span_ of an interval \(I_{u}\) is given by \(\delta(I_{u})=e_{u}-s_{u}\). We say that a timeline \(\mathcal{T}=\{I_{u}\}_{u\in V}\) covers a temporal graph \(G=(V,E)\) if for every edge \((u,v,t)\in E\), the timestamp \(t\) belongs to either \(I_{u}\) or \(I_{v}\). A naive timeline, defined as \(I_{u}=[\min T(u),\max T(u)]\), provides full coverage but may include unnecessarily extended intervals. The objective is thus to determine a timeline minimizing the sum-span, expressed as \(S(\mathcal{T})=\sum_{u\in V}\delta(I_{u})\). **Problem 1** (MinTcover\({}_{+}\)): _Given a temporal graph \(G=(V,E)\), we identify a timeline \(\mathcal{T}=\{I_{u}\}_{u\in V}\) that covers \(G\) while minimizing the sum-span \(S(\mathcal{T})\)._  
With the above definition, we can state our goal, which requires us to define the problem as follows. Given a temporal graph \(G=(V,E)\), where \(V\) is the set of vertices and \(E\) is the set of temporal edges, we find a parametric function \(f\) that maps the input temporal graph \(G\) to the set of activity intervals \(\mathcal{T}=\{I_{u}\}_{u\in V}\), denoted as \(f(G;\theta)=\mathcal{T}\), where \(\theta\) represents the parameters of the red function. Suppose each vertex \(u\) can be represented at each timestamp \(t\) where it is active as a feature vector \(P_{u}^{t}\) through a parametric function \(g\). Let us suppose these representationscapture both the temporal and structural properties of the graph relevant to determine the activity intervals, considering the fact that if an interval \(I_{v}\) for node \(v\in V\) is part of the activity timeline it influences, the interval \(I_{u}\) for node \(u\) should be part of the activity timeline if \(u\) and \(v\) are adjacent in some timestamp. Thus, for each node, we have a sequence of representations for each timestamp. Let us call these sequences \(\mathcal{P}=\{P^{t}_{u}\}_{u\in V,t\in\mathcal{T}}\) and \(g(G;\theta_{1})=\mathcal{P}\), the parametric function that maps the input graph \(G\) to its representation \(\mathcal{P}\), with \(\theta_{1}\) parameters of the function. For each of these sequences, we want to model the probability that a certain node \(u\) is active at time \(t\), resulting in an interval \(I_{u}\) that belongs to an activity timeline \(\mathcal{T}=\{I_{u}\}_{u\in V}\). Thus, we need to define another parametric function \(f(\mathcal{P};\theta_{2})=\mathcal{T}\), which predicts the activity interval \([s_{u},e_{u}]\) of each node \(u\), using the learned parameters \(\theta_{2}\). We reformulate the problem as follows. **Problem 2** (Minimum Timeline Cover with Least Squares).