We examine the problem of maximizing the reachability of a given source in temporal graphs that are given as the union of k temporal paths, i.e., every given path is a sequence of edges with strictly increasing labels that denote availability in time. This type of temporal graphs represent train networks. We consider shifting operations on the labels of the paths that maintain their temporal continuity. This means that we can move the availability of a temporal edge later or earlier in time, and propagate the shifts to all other affected edges of the path in order to preserve its temporal connectivity. We study the parameterized complexity of the problem with respect to the number of paths k, and the total budget b, where b is the maximum number of shifts we are allowed to perform. Our results reveal that fixed parameter tractability can be achieved (1) when parameterized both by k and b, and (2) when parameterized by k, and b is unlimited. In almost every other case, e.g., parameterized by a single parameter or parameterized by k, while having a bound on b, we establish intractability lower bounds that are matched by XP algorithms.

The Temporal Graph Exploration problem (TEXP) takes as input a temporal graph, i.e., a sequence of graphs \((G_i)_{i\in\mathbb N}\) on the same vertex set, and asks for a walk of shortest length visiting all vertices, where the \(i\)-th step uses an edge from \(G_i\) or stays put. If each such \(G_i\) is connected, then an exploration of length \(n^2\) exists, and this is known to be the best possible up to a constant. More fine-grained lower and upper bounds have been obtained for restricted temporal graph classes, however, for several fundamental classes, a large gap persists between known bounds, and it remains unclear which properties of a temporal graph make it inherently difficult to explore.

Logistics and transportation networks require a large amount of resources to realise necessary connections between locations and minimizing these resources is a vital aspect of planning research. Since such networks have dynamic connections that are only available at specific times, intricate models are needed to portray them accurately. In this paper, we study the problem of minimizing the number of resources needed to realise a dynamic network, using the temporal graphs model. In a temporal graph, edges appear at specific points in time. Given a temporal graph and a natural number k, we ask whether we can cover every temporal edge exactly once using at most k temporal journeys; in a temporal journey consecutive edges have to adhere to the order of time. We conduct a thorough investigation of the complexity of the problem with respect to four dimensions: (a) whether the type of the temporal journey is a walk, a trail, or a path; (b) whether the chronological order of edges in the journey is strict or non-strict; (c) whether the temporal graph is directed or undirected; (d) whether the start and end points of each journey are given. We almost completely resolve the complexity of these problems and provide dichotomies for each of them with respect to k.

Researchers, policy makers, and engineers need to make sense of data from spreading processes as diverse as rumor spreading in social networks, viral infections, and water contamination. Classical questions include predicting infection behavior in a given network or deducing the network structure from infection data. Most of the research on network infections studies static graphs, that is, the connections in the network are assumed to not change. More recently, temporal graphs, in which connections change over time, have been used to more accurately represent real-world infections, which rarely occur in unchanging networks. We propose a model for temporal graph discovery that is consistent with previous work on static graphs and embraces the greater expressiveness of temporal graphs. For this model, we give algorithms and lower bounds which are often tight. We analyze different variations of the problem, which make our results widely applicable and it also clarifies which aspects of temporal infections make graph discovery easier or harder. We round off our analysis with an experimental evaluation of our algorithm on real-world interaction data from the Stanford Network Analysis Project and on temporal Erdős-Renyi graphs. On Erdős-Renyi graphs, we uncover a threshold behavior, which can be explained by a novel connectivity parameter that we introduce during our theoretical analysis.

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from (2/√3) to 4/3 depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

We study a new algorithmic process of graph growth which starts from a single initial vertex and operates in discrete time-steps, called slots . In every slot, the graph grows via two operations (i) vertex generation and (ii) edge activation. The process completes at the last slot where a (possibly empty) subset of the edges of the graph are removed. Removed edges are called excess edges . The main problem investigated in this paper is: Given a target graph G , design an algorithm that outputs a process that grows G , called a growth schedule . Additionally, we aim to minimize the total number of slots k and of excess edges ℓ used by the process. We provide both positive and negative results, with our main focus being either schedules with sub-linear number of slots or with no excess edges.

In this paper, we study the main open question of [Michail, Skretas, Spirakis, ICALP'17], asking what are the families of two-dimensional geometric shapes, drawn on a square grid, that can be transformed into each other by a sequence of rotation operations, none of which disconnects the shape. The model represents programmable matter systems consisting of interconnected robotic modules that perform the minimal mechanical operation of 90° rotations around each other. The goal is to transform an initial connected shape of modules A into a target connected shape B . Under the necessary assumption that the two shapes have identical colour cardinalities on a checkered colouring of the grid, and using at most a constant number of auxiliary modules to trigger the transformation, we prove that almost any pair of such shapes can be transformed into each other within an optimal O ( n 2 ) rotation operations none of which disconnects the shape.

Many real-world networks, such as transportation or trade networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners in temporal cliques, i.e., sparse subgraphs of complete temporal graphs that ensure all-pairs reachability via temporal paths. So far, the best known result is the existence of temporal spanners with $\mathcal{O}(n\log n)$ many edges. We present significant progress towards proving that linear-size temporal spanners exist in all temporal cliques. We adapt techniques used in previous works and heavily expand and generalize them to provide a simpler and more intuitive proof of the $\mathcal{O}(n\log n)$ bound. Moreover, we use our novel approach to show that a large class of temporal cliques, called edge-pivot graphs, admit linear-size temporal spanners. To contrast this, we investigate other classes of temporal cliques that do not belong to the class of edge-pivot graphs. We introduce two such graph classes and we develop novel techniques for establishing the existence of linear temporal spanners in these graph classes as well.

Most networks are not static objects, but instead they change over time. This observation has sparked rigorous research on temporal graphs within the last years. In temporal graphs, we have a fixed set of nodes and the connections between them are only available at certain time steps. This gives rise to a plethora of algorithmic problems on such graphs, most prominently the problem of finding temporal spanners, i.e., the computation of subgraphs that guarantee all pairs reachability via temporal paths. To the best of our knowledge, only centralized approaches for the solution of this problem are known. However, many real-world networks are not shaped by a central designer but instead they emerge and evolve by the interaction of many strategic agents. This observation is the driving force of the recent intensive research on game-theoretic network formation models. In this work we bring together these two recent research directions: temporal graphs and game-theoretic network formation. As a first step into this new realm, we focus on a simplified setting where a complete temporal host graph is given and the agents, corresponding to its nodes, selfishly create incident edges to ensure that they can reach all other nodes via temporal paths in the created network. This yields temporal spanners as equilibria of our game. We prove results on the convergence to and the existence of equilibrium networks, on the complexity of finding best agent strategies, and on the quality of the equilibria. By taking these first important steps, we uncover challenging open problems that call for an in-depth exploration of the creation of temporal graphs by strategic agents.

We study how we can accelerate the spreading of information in temporal graphs via shifting operations; a problem that captures real-world applications varying from information flows to distribution schedules. In a temporal graph there is a set of fixed vertices and the available connections between them change over time in a predefined manner. We observe that, in some cases, shifting some connections, i.e., advancing or delaying them, can decrease the time required to reach from some vertex (source) to another vertex. We study how we can minimize the maximum time a set of sources needs to reach every vertex, when we are allowed to shift some of the connections. If we restrict the allowed number of changes, we prove that, already for a single source, the problem is NP-hard, and W[2]-hard when parameterized by the number of changes. Then we focus on unconstrained number of changes. We derive a polynomial-time algorithm when there is one source. When there are two sources, we show that the problem becomes NP-hard; on the other hand, we design an FPT algorithm parameterized by the treewidth of the graph plus the lifetime of the optimal solution, that works for any number of sources. Finally, we provide polynomial-time algorithms for several graph classes.

Abstract We study here systems of distributed entities that can actively modify their communication network. This gives rise to distributed algorithms that apart from communication can also exploit network reconfiguration to carry out a given task. Also, the distributed task itself may now require a global reconfiguration from a given initial network $$G_s$$ G s to a target network $$G_f$$ G f from a desirable family of networks. To formally capture costs associated with creating and maintaining connections, we define three edge-complexity measures: the total edge activations , the maximum activated edges per round , and the maximum activated degree of a node . We give (poly)log( n ) time algorithms for the task of transforming any $$G_s$$ G s into a $$G_f$$ G f of diameter (poly)log( n ), while minimizing the edge-complexity. Our main lower bound shows that $$\varOmega (n)$$ Ω ( n ) total edge activations and $$\varOmega (n/\log n)$$ Ω ( n / log n ) activations per round must be paid by any algorithm (even centralized) that achieves an optimum of $$\varTheta (\log n)$$ Θ ( log n ) rounds. We give three distributed algorithms for our general task. The first runs in $$O(\log n)$$ O ( log n ) time, with at most 2 n active edges per round, a total of $$O(n\log n)$$ O ( n log n ) edge activations, a maximum degree $$n-1$$ n - 1 , and a target network of diameter 2. The second achieves bounded degree by paying an additional logarithmic factor in time and in total edge activations. It gives a target network of diameter $$O(\log n)$$ O ( log n ) and uses O ( n ) active edges per round. Our third algorithm shows that if we slightly increase the maximum degree to polylog( n ) then we can achieve $$o(\log ^2 n)$$ o ( log 2 n ) running time. </

In this paper, we study systems of distributed entities that can actively modify their communication network. This gives rise to distributed algorithms that apart from communication can also exploit network reconfiguration in order to carry out a given task. At the same time, the distributed task itself may now require a global reconfiguration from a given initial network Gs to a target network Gf from a family of networks having some good properties, like small diameter.

In this work, we study theoretical models of programmable matter systems. The systems under consideration consist of spherical modules, kept together by magnetic forces and able to perform two minimal mechanical operations (or movements): rotate around a neighbor and slide over a line. In terms of modeling, there are n nodes arranged in a 2-dimensional grid and forming some initial shape. The goal is for the initial shape A to transform to some target shape B by a sequence of movements. Most of the paper focuses on transformability questions, meaning whether it is in principle feasible to transform a given shape to another. We first consider the case in which only rotation is available to the nodes. Our main result is that deciding whether two given shapes A and B can be transformed to each other is in P. We then insist on rotation only and impose the restriction that the nodes must maintain global connectivity throughout the transformation. We prove that the corresponding transformability question is in PSPACE and study the problem of determining the minimum seeds that can make feasible otherwise infeasible transformations. Next we allow both rotations and slidings and prove universality: any two connected shapes A,B of the same number of nodes, can be transformed to each other without breaking connectivity. The worst-case number of movements of the generic strategy is Theta(n^2). We improve this to O(n) parallel time, by a pipelining strategy, and prove optimality of both by matching lower bounds. We next turn our attention to distributed transformations. The nodes are now distributed processes able to perform communicate-compute-move rounds. We provide distributed algorithms for a general type of transformation.