Stefano Leucci’s research while affiliated with University of L'Aquila and other places

In a temporal forest each edge has an associated set of time labels that specify the time instants in which the edges are available. A temporal path from vertex u to vertex v in the forest is a selection of a label for each edge in the unique path from u to v, assuming it exists, such that the labels selected for any two consecutive edges are non-decreasing. We design linear-size data structures that maintain a temporal forest of rooted trees under addition and deletion of both edge labels and singleton vertices, insertion of root-to-node edges, and removal of edges with no labels. Such data structures can answer temporal reachability, earliest arrival, and latest departure queries. All queries and updates are handled in polylogarithmic worst-case time. Our results can be adapted to deal with latencies. More precisely, all the worst-case time bounds are asymptotically unaffected when latencies are uniform. For arbitrary latencies, the update time becomes amortized in the incremental case where only label additions and edge/singleton insertions are allowed as well as in the decremental case in which only label deletions and edge/singleton removals are allowed. To the best of our knowledge, the only previously known data structure supporting temporal reachability queries is due to Brito, Albertini, Casteigts, and Traven\c{c}olo [Social Network Analysis and Mining, 2021], which can handle general temporal graphs, answers queries in logarithmic time in the worst case, but requires an amortized update time that is quadratic in the number of vertices, up to polylogarithmic factors.

A spanner is a sparse subgraph of a given graph G which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t.\ the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph G of n vertices asks to find the largest set of vertices $X\subseteq V(G)$, also called $\mu$-set, such that for any two vertices $u,v\in X$, there is a shortest u,v-path P where all internal vertices of P are not in X. This means that u and v are visible w.r.t. X. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside X. The mutual-visibility problem and all its variations are known to be $\mathsf{NP}$-complete on graphs of diameter 4. In this paper, we design a polynomial-time algorithm that finds a $\mu$-set with size $\Omega\left( \sqrt{n/ \overline{D}} \right)$, where $\overline D$ is the average distance between any two vertices of G. Moreover, we show inapproximability results for all visibility problems on graphs of diameter 2 and strengthen the inapproximability ratios for graphs of diameter 3 or larger. More precisely, for graphs of diameter at least 3 and for every constant $\varepsilon > 0$, we show that mutual-visibility and dual mutual-visibility problems are not approximable within a factor of $n^{1/3-\varepsilon}$, while outer and total mutual-visibility problems are not approximable within a factor of $n^{1/2 - \varepsilon}$, unless $\mathsf{P}=\mathsf{NP}$. Furthermore we study the relationship between the mutual-visibility number and the general position number in which no three distinct vertices u,v,w of X belong to any shortest path of G.

Given an undirected connected graph $G = (V(G), E(G))$ on n vertices, the minimum Monitoring Edge-Geodetic Set (MEG-set) problem asks to find a subset $M \subseteq V(G)$ of minimum cardinality such that, for every edge $e \in E(G)$, there exist $x,y \in M$ for which all shortest paths between x and y in G traverse e. We show that, for any constant $c < \frac{1}{2}$, no polynomial-time $(c \log n)$-approximation algorithm for the minimum MEG-set problem exists, unless $\mathsf{P} = \mathsf{NP}$.

In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices as input. The tree is embedded in an unknown metric space and we have unlimited access to an oracle that, given two distinct vertices u and v of T, can answer queries reporting the cost of the edge (u, v) in constant time. We want to augment T with k shortcuts in order to minimize the diameter of the resulting graph.For k=1, $O(n \log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bilò, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than 10/9-approximate solution for trees for $k\ge 3$. For any constant $\varepsilon > 0$, we instead design a linear-time $(1+\varepsilon )$-approximation algorithm for paths and $k = o(\sqrt{\log n})$, thus establishing a dichotomy between paths and trees for $k\ge 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time 4-approximation algorithm for trees, and to compute the diameter of graphs with $n \,+\, k\, -\, 1$ edges in time $O(n k \log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.KeywordsTree diameter augmentationFast diameter computationApproximation algorithmsTime-efficient algorithms

In the \emph{k-Diameter-Optimally Augmenting Tree Problem} we are given a tree T of n vertices as input. The tree is embedded in an unknown \emph{metric} space and we have unlimited access to an oracle that, given two distinct vertices u and v of T, can answer queries reporting the cost of the edge (u,v) in constant time. We want to augment T with k shortcuts in order to minimize the diameter of the resulting graph. For k=1, $O(n \log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bil\`o, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than 10/9-approximate solution for trees for $k\geq 3$. For any constant $\varepsilon > 0$, we instead design a linear-time $(1+\varepsilon)$-approximation algorithm for paths and $k = o(\sqrt{\log n})$, thus establishing a dichotomy between paths and trees for $k\geq 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time 4-approximation algorithm for trees, and to compute the diameter of graphs with $n + k - 1$ edges in time $O(n k \log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.

In this paper we introduce the notions of blackout-tolerant temporal $\alpha$-spanner of a temporal graph G which is a subgraph of G that preserves the distances between pairs of vertices of interest in G up to a multiplicative factor of $\alpha$, even when the graph edges at a single time-instant become unavailable. In particular, we consider the single-source, single-pair, and all-pairs cases and, for each case we look at three quality requirements: exact distances (i.e., $\alpha =1$), almost-exact distances (i.e., $\alpha = 1+\varepsilon$ for an arbitrarily small constant $\varepsilon >0$), and connectivity (i.e., unbounded $\alpha$). For each combination we provide tight bounds, up to polylogarithmic factors, on the size, which is measured as the number of edges, of the corresponding blackout-tolerant $\alpha$-spanner for both general temporal graphs and for temporal cliques. Our result show that such spanners are either very sparse (i.e., they have $\widetilde{O}(n)$ edges) or they must have size $\varOmega (n^2)$ in the worst case, where n is the number of vertices of G. To complete the picture, we also investigate the case of multiple blackouts.KeywordsTemporal graphsTemporal spannersFault-tolerance

We study the problem of designing a resilient data structure maintaining a tree under the Faulty-RAM model [Finocchi and Italiano, STOC’04] in which up to $\delta$ δ memory words can be corrupted by an adversary. Our data structure stores a rooted dynamic tree that can be updated via the addition of new leaves, requires linear size, and supports resilient (weighted) level ancestor queries, lowest common ancestor queries, and bottleneck vertex queries in $O(\delta )$ O ( δ ) worst-case time per operation.

We study the heterogeneous weighted delivery (HWD) problem introduced in [Bärtschi et al., STACS'17] where k heterogeneous mobile agents (e.g., robots, vehicles, etc.), initially positioned on vertices of an n-vertex edge-weighted graph G, have to deliver m messages. Each message is initially placed on a source vertex of G and needs to be delivered to a target vertex of G. Each agent can move along the edges of G and carry at most one message at any time. Each agent has a rate of energy consumption per unit of traveled distance and the goal is that of delivering all messages using the minimum overall amount of energy. This problem has been shown to be NP-hard even when k=1, and is 4ρ-approximable where ρ is the ratio between the maximum and minimum energy consumption of the agents. In this paper, we provide approximation algorithms with approximation ratios independent of the energy consumption rates. First, we design a polynomial-time 8-approximation algorithm for k=O(log⁡n), closing a problem left open in [Bärtschi et al., ATMOS'17]. This algorithm can be turned into an O(k)-approximation algorithm that always runs in polynomial-time, regardless of the values of k. Then, we show that HWD problem is 36-approximable in polynomial-time when each agent has one of two possible consumption rates. Finally, we design a polynomial-time O˜(log3⁡n)-approximation algorithm for the general case.