arXiv subjects

Filippo Brunelli

Publications and source records attributed to Filippo Brunelli.

6 recordsLinked to original sources

arXiv2025

Making Temporal Betweenness Computation Faster and Restless

Bu{\ss} et al [KDD 2020] recently proved that the problem of computing the betweenness of all nodes of a temporal graph is computationally hard in the case of foremost and fastest paths, while it is solvable in time O(n 3 T 2 ) in the case of shortest and shortest foremost paths, where n is the number of nodes and T is the number of distinct time steps. A new algorithm for temporal betweenness computation is introduced in this paper. In the case of shortest and shortest foremost paths, it requires O(n + M ) space and runs in time where M is the number of temporal edges, thus significantly improving the algorithm of Bu{\ss} et al in terms of time complexity (note that T is usually large). Experimental evidence is provided that our algorithm performs between twice and almost 250 times better than the algorithm of Bu{\ss} et al. Moreover, we were able to compute the exact temporal betweenness values of several large temporal graphs with over a million of temporal edges. For such size, only approximate computation was possible by using the algorithm of Santoro and Sarpe [WWW 2022]. Maybe more importantly, our algorithm extends to the case of restless walks (that is, walks with waiting constraints in each node), thus providing a polynomial-time algorithm (with complexity O(nM )) for computing the temporal betweenness in the case of several different optimality criteria. Such restless computation was known only for the shortest criterion (Rymar et al [JGAA 2023]), with complexity O(n 2 M T 2 ). We performed an extensive experimental validation by comparing different waiting constraints and different optimisation criteria. Moreover, as a case study, we investigate six public transit networks including Berlin, Rome, and Paris. Overall we find a general consistency between the different variants of betweenness centrality. However, we do measure a sensible influence of waiting constraints, and note some cases of low correlation for certain pairs of criteria in some networks.

cs.DS↗

arXiv2025

Complexity Gaps between Point and Interval Temporal Graphs for some Reachability Problems

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration, delay of transmission. In the point model, edges appear at specific points in time, whereas in the more general interval model, edges can be present over specific time intervals. In both models, the delay for traversing an edge can change with each edge appearance. When time is discrete, the two models are equivalent in the sense that the presence of an edge during an interval is equivalent to a sequence of point-in-time occurrences of the edge. However, this transformation can drastically change the size of the input and has implications for complexity. Indeed, we show a gap between the two models with respect to the complexity of the classical problem of computing a fastest temporal path from a source vertex to a target vertex, i.e. a path where edges can be traversed one after another in time and such that the total duration from source to target is minimized. It can be solved in near-linear time in the point model, while we show that the interval model requires quadratic time under classical assumptions of fine-grained complexity. With respect to linear time, our lower bound implies a factor of the number of vertices, while the best known algorithm has a factor of the number of underlying edges. We also show a similar complexity gap for computing a shortest temporal path, i.e. a temporal path with a minimum number of edges. Here our lower bound matches known upper bounds up to a logarithmic factor. Interestingly, we show that near-linear time for fastest temporal path computation is possible in the interval model when it is restricted to uniform delay zero, i.e., when traversing an edge is instantaneous. However, this special case is not exempt from our lower bound for shortest temporal path computation. These two results should be contrasted with the computation of a foremost temporal path, i.e., a temporal path that arrives as early as possible. It is well known that this computation can be solved in near-linear time in both models. We also show that there is no gap in testing the all-to-all temporal connectivity of a temporal graph. We demonstrate a quadratic lower bound that applies to both the interval and point models and aligns with the existing upper bounds.

cs.DS↗

arXiv2023

A Note on the Complexity of Maximizing Temporal Reachability via Edge Temporalisation of Directed Graphs

A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of assigning time labels to the edges of a digraph in order to maximize the total reachability of the resulting temporal graph (that is, the number of pairs of nodes which are connected one to the other). In particular, we prove that this problem is NP-hard. We then conjecture that the problem is approximable within a constant approximation ratio. This conjecture is a consequence of the following graph theoretic conjecture: any strongly connected directed graph with n nodes admits an out-arborescence and an in-arborescence that are edge-disjoint, have the same root, and each spans $\Omega$(n) nodes.

cs.DS↗

arXiv2022

Minimum-Cost Temporal Walks under Waiting-Time Constraints in Linear Time

In a temporal graph, each edge is available at specific points in time. Such an availability point is often represented by a ''temporal edge'' that can be traversed from its tail only at a specific departure time, for arriving in its head after a specific travel time. In such a graph, the connectivity from one node to another is naturally captured by the existence of a temporal path where temporal edges can be traversed one after the other. When imposing constraints on how much time it is possible to wait at a node in-between two temporal edges, it then becomes interesting to consider temporal walks where it is allowed to visit several times the same node, possibly at different times. We study the complexity of computing minimum-cost temporal walks from a single source under waiting-time constraints in a temporal graph, and ask under which conditions this problem can be solved in linear time. Our main result is a linear time algorithm when the input temporal graph is given by its (classical) space-time representation. We use an algebraic framework for manipulating abstract costs, enabling the optimization of a large variety of criteria or even combinations of these. It allows to improve previous results for several criteria such as number of edges or overall waiting time even without waiting constraints. It saves a logarithmic factor for all criteria under waiting constraints. Interestingly, we show that a logarithmic factor in the time complexity appears to be necessary with a more basic input consisting of a single ordered list of temporal edges (sorted either by arrival times or departure times). We indeed show equivalence between the space-time representation and a representation with two ordered lists.

cs.DS↗

arXiv2021

On The Complexity of Maximizing Temporal Reachability via Trip Temporalisation

We consider the problem of assigning appearing times to the edges of a digraph in order to maximize the (average) temporal reachability between pairs of nodes. Motivated by the application to public transit networks, where edges cannot be scheduled independently one of another, we consider the setting where the edges are grouped into certain walks (called trips) in the digraph and where assigning the appearing time to the first edge of a trip forces the appearing times of the subsequent edges. In this setting, we show that, quite surprisingly, it is NP-complete to decide whether there exists an assignment of times connecting a given pair of nodes. This result allows us to prove that the problem of maximising the temporal reachability cannot be approximated within a factor better than some polynomial term in the size of the graph. We thus focus on the case where, for each pair of nodes, there exists an assignment of times such that one node is reachable from the other. We call this property strong temporalisability. It is a very natural assumption for the application to public transit networks. On the negative side, the problem of maximising the temporal reachability remains hard to approximate within a factor $\sqrt$ n/12 in that setting. Moreover, we show the existence of collections of trips that are strongly temporalisable but for which any assignment of starting times to the trips connects at most an O(1/ $\sqrt$ n) fraction of all pairs of nodes. On the positive side, we show that there must exist an assignment of times that connects a constant fraction of all pairs in the strongly temporalisable and symmetric case, that is, when the set of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order. Keywords:edge labeling edge scheduled network network optimisation temporal graph temporal path temporal reachability time assignment

cs.DM↗

arXiv2021

On Computing Pareto Optimal Paths in Weighted Time-Dependent Networks

A weighted point-availability time-dependent network is a list of temporal edges, where each temporal edge has an appearing time value, a travel time value, and a cost value. In this paper we consider the single source Pareto problem in weighted point-availability time-dependent networks, which consists of computing, for any destination d, all Pareto optimal pairs (t, c), where t and c are the arrival time and the cost of a path from s to d, respectively (a pair (t, c) is Pareto optimal if there is no path with arrival time smaller than t and cost no worse than c or arrival time no greater than t and better cost). We design and analyse a general algorithm for solving this problem, whose time complexity is O(M log P), where M is the number of temporal edges and P is the maximum number of Pareto optimal pairs for each node of the network. This complexity significantly improves the time complexity of the previously known solution. Our algorithm can be used to solve several different minimum cost path problems in weighted point-availability time-dependent networks with a vast variety of cost definitions, and it can be easily modified in order to deal with the single destination Pareto problem. All our results apply to directed networks, but they can be easily adapted to undirected networks with no edges with zero travel time.

cs.DS↗