Consider planning a trip in a train network. In contrast to, say, a road network, the edges are temporal, i.e., they are only available at certain times. Another important difficulty is that trains, unfortunately, sometimes get delayed. This is especially bad if it causes one to miss subsequent trains. The best way to prepare against this is to have a connection that is robust to some number of (small) delays. An important factor in determining the robustness of a connection is how far in advance delays are announced. We give polynomial-time algorithms for the two extreme cases: delays known before departure and delays occurring without prior warning (the latter leading to a two-player game scenario). Interestingly, in the latter case, we show that the problem becomes PSPACE-complete if the itinerary is demanded to be a path.

Abstract. A temporal graph is a graph whose edges appear only at certain points in time. Recently, the second and the last three authors proposed a natural temporal analog of the Erdős–Rényi random graph model. The proposed model is obtained by randomly permuting the edges of an Erdős–Rényi random graph and interpreting this permutation as an ordering of presence times. It was shown that the connectivity threshold in the Erdős–Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting. In the present paper, we identify a sharp threshold for the emergence of a giant temporally connected component. We show that at [Formula: see text] the size of the largest temporally connected component increases from [Formula: see text] to [Formula: see text]. This threshold holds for both open and closed connected components, i.e., components that allow (respectively, forbid) their connecting paths to use external nodes.

In a temporal network with discrete time-labels on its edges, information can only "flow" along sequences of edges with non-decreasing (resp.increasing) time-labels.In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges.By naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation, and we systematically investigate its algorithmic behavior.Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a temporal graph G is transitively orientable.In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether G is strictly transitively orientable.Additionally we introduce further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

We introduce the algorithmic problem of finding a locally rainbow path of length l connecting two distinguished vertices s and t in a vertex-colored directed graph. Herein, a path is locally rainbow if between any two visits of equally colored vertices, the path traverses consecutively at leaset r differently colored vertices. This problem generalizes the well-known problem of finding a rainbow path. It finds natural applications whenever there are different types of resources that must be protected from overuse, such as crop sequence optimization or production process scheduling. We show that the problem is computationally intractable even if r=2 or if one looks for a locally rainbow among the shortest paths. On the positive side, if one looks for a path that takes only a short detour (i.e., it is slightly longer than the shortest path) and if r is small, the problem can be solved efficiently. Indeed, the running time of the respective algorithm is near-optimal unless the ETH fails.

Abstract. A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erdős–Rényi random graph, [Formula: see text], by considering a random permutation [Formula: see text] of the edges and interpreting the ranks in [Formula: see text] as presence times. We give a thorough study of the temporal connectivity of such graphs and derive implications for the existence of several kinds of sparse spanners. It turns out that temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that at [Formula: see text], any fixed pair of vertices can asymptotically almost surely (a.a.s.) reach each other; at [Formula: see text], at least one vertex (and, in fact, any fixed vertex) can a.a.s. reach all others; and at [Formula: see text], all the vertices can a.a.s. reach each other; i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size [Formula: see text] as soon as it becomes temporally connected, which is nearly optimal, as [Formula: see text] is a lower bound. This result is quite significant because temporal graphs do not admit spanners of size [Formula: see text] in general [Kempe, Kleinberg, and Kumar, J. Comput. System Sci., 64 (2002), pp. 820–842]. In fact, they do not even always admit spanners of size [Formula: see text] [Axiotis and Fotakis, On the size and the approximability of minimum temporally connected subgraphs, 2016, pp. 149:1–149:14]. Thus, our result implies that the obstructions found in these works—and more generally any non-negligible obstruction—are statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners a step further, we show that pivotal spanners—i.e., spanners of size [Formula: see text] composed of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently)—exist a.a.s. at [Formula: see text], this threshold being also sharp. Finally, we show that optimal spanners (of size [Formula: see text]) also exist a.a.s. at [Formula: see text]. Whether this value is a sharp threshold is open; we conjecture that it is. For completeness, we compare the above results to existing results in related areas, including edge-ordered graphs, gossip theory, and population protocols, showing that our results can be interpreted in these settings as well and that in some cases they improve known results therein. Finally, we discuss an intriguing connection between our results and Janson’s celebrated results on percolation in weighted graphs.

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network $N$ can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of $N$ and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

Abstract. Given an undirected [Formula: see text]-vertex graph and [Formula: see text] pairs [Formula: see text] of terminal vertices, the [Formula: see text]-Disjoint Shortest Paths ([Formula: see text]-SDP) problem asks whether there are [Formula: see text] pairwise vertex-disjoint paths [Formula: see text] such that [Formula: see text] is a shortest [Formula: see text]-[Formula: see text]-path for each [Formula: see text]. Recently, Lochet [ Proceedings of the 32 nd ACM-SIAM Symposium on Discrete Algorithms (SODA ’21 ), SIAM, 2021, pp. 169–178] provided an algorithm that solves [Formula: see text]-SDP in [Formula: see text] time, answering a 20-year old question about the computational complexity of [Formula: see text]-SDP for constant [Formula: see text]. On the one hand, we present an improved [Formula: see text]-time algorithm based on a novel geometric view on this problem. For the special case [Formula: see text] on [Formula: see text]-edge graphs, we show that the running time can be further reduced to [Formula: see text] by small modifications of the algorithm and a refined analysis. On the other hand, we show that [Formula: see text]-SDP is W[1]-hard with respect to [Formula: see text], showing that the dependency of the degree of the polynomial running time on the parameter [Formula: see text] is presumably unavoidable.

Most transportation networks are inherently temporal: Connections (e.g. flights, train runs) are only available at certain, scheduled times. When transporting passengers or commodities, this fact must be considered for the the planning of itineraries. This has already led to several well-studied algorithmic problems on temporal graphs. The difficulty of the described task is increased by the fact that connections are often unreliable - in particular, many modes of transportation suffer from occasional delays. If these delays cause subsequent connections to be missed, the consequences can be severe. Thus, it is a vital problem to design itineraries that are robust to (small) delays. We initiate the study of this problem from a parameterized complexity perspective by proving its NP-completeness as well as several hardness and tractability results for natural parameterizations.

The world is rarely static - many problems need not only be solved once but repeatedly, under changing conditions. This setting is addressed by the multistage view on computational problems. We study the diverse multistage variant, where consecutive solutions of large variety are preferable to similar ones, e.g. for reasons of fairness or wear minimization. While some aspects of this model have been tackled before, we introduce a framework allowing us to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. This is achieved by first solving special, colored variants of these problems, which might also be of independent interest.

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erdös-Rényi random graph G ~ Gn,p by considering a random permutation π of the edges and interpreting the ranks in π as presence times. We give a thorough study of the temporal connectivity of such graphs and derive implications for the existence of several kinds of sparse spanners. It turns out that temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that, at p = log$n$/n, any fixed pair of vertices can a.a.s. reach each other; at 2 log$n$/n, at least one vertex (and in fact, any fixed vertex) can a.a.s. reach all others; and at 3 log$n$/n, all the vertices can a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size 2n + o(n) as soon as it becomes temporally connected, which is nearly optimal as 2n - 4 is a lower bound. This result is quite significant because temporal graphs do not admit spanners of size O(n) in general (Kempe, Kleinberg, Kumar, STOC 2000). In fact, they do not even always admit spanners of size o($n$2) (Axiotis, Fotakis, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, any non-negligible obstruction is statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners a step further, we show that pivotal spanners-i.e., spanners of size 2n - 2 made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently)-exist a.a.s. at 4 log$n$/ n, this threshold being also sharp. Finally, we show that optimal spanners (of size 2n - 4) also exist a.a.s. at p = 4 log$n$/n, Whether this value is a sharp threshold is open, we conjecture that it is. For completeness, we compare the above results to existing results in related areas, including edge-ordered graphs, gossip theory, and population protocols, showing that our results can be interpreted in these settings as well, and that in some cases, they improve known results therein. Finally, we discuss an intriguing connection between our results and Janson's celebrated results on percolation in weighted graphs.

Given an undirected $n$-vertex graph and $k$ pairs of terminal vertices $(s_1,t_1), \ldots, (s_k,t_k)$, the $k$-Disjoint Shortest Paths ($k$-DSP)-problem asks whether there are $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ is a shortest $s_i$-$t_i$-path for each $i \in [k]$. Recently, Lochet [SODA 2021] provided an algorithm that solves $k$-DSP in $n^{O(k^{5^k})}$ time, answering a 20-year old question about the computational complexity of $k$-DSP for constant $k$. On the one hand, we present an improved $n^{O(k!k)}$-time algorithm based on a novel geometric view on this problem. For the special case $k=2$ on $m$-edge graphs, we show that the running time can be further reduced to $O(nm)$ by small modifications of the algorithm and a refined analysis. On the other hand, we show that $k$-DSP is W[1]-hard with respect to $k$, showing that the dependency of the degree of the polynomial running time on the parameter $k$ is presumably unavoidable.

To address the dynamic nature of real-world networks, we generalize competitive diffusion games and Voronoi games from static to temporal graphs, where edges may appear or disappear over time. This establishes a new direction of studies in the area of graph games, motivated by applications such as influence spreading. As a first step, we investigate the existence of Nash equilibria in competitive diffusion and Voronoi games on different temporal graph classes. Even when restricting our studies to temporal paths and cycles, this turns out to be a challenging undertaking, revealing significant differences between the two games in the temporal setting. Notably, both games are equivalent on static paths and cycles. Our two main technical results are (algorithmic) proofs for the existence of Nash equilibria in temporal competitive diffusion and temporal Voronoi games when the edges are restricted not to disappear over time.

In a temporal network with discrete time-labels on its edges, entities and information can only "flow" along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge e = {u,v} with time-label t specifies that "u communicates with v at time t". This is a symmetric relation between u and v, and it can be interpreted that the information can flow in either direction. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior in various situations. An orientation of a temporal graph is called temporally transitive if, whenever u has a directed edge towards v with time-label t₁ and v has a directed edge towards w with time-label t₂ ≥ t₁, then u also has a directed edge towards w with some time-label t₃ ≥ t₂. If we just demand that this implication holds whenever t₂ > t₁, the orientation is called strictly temporally transitive, as it is based on the fact that there is a strict directed temporal path from u to w. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph 𝒢 is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether 𝒢 is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

We study spreading processes in temporal graphs, i. e., graphs whose connections change over time. These processes naturally model real-world phenomena such as infectious diseases or information flows. More precisely, we investigate how such a spreading process, emerging from a given set of sources, can be contained to a small part of the graph. To this end we consider two ways of modifying the graph, which are (1) deleting connections and (2) delaying connections. We show a close relationship between the two associated problems and give a polynomial time algorithm when the graph has tree structure. For the general version, we consider parameterization by the number of vertices to which the spread is contained. Surprisingly, we prove W[1]-hardness for the deletion variant but fixed-parameter tractability for the delaying variant.

Abstract Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications in areas such as physics, geography, and medical sciences. Computing shortest paths in visibility graphs is a common task, for example, in line-of-sight communication. For time series analysis, graph characteristics involving shortest paths lengths (such as centrality measures) have proven useful. In this paper, we devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visibility graphs). Our algorithm runs in $$O(d^*\log \varDelta )$$ O ( d ∗ log Δ ) time, where $$d^*$$ d ∗ is the length (that is, the number of edges) of the shortest path and $$\varDelta $$ Δ is the maximum vertex degree. Alternatively, with an $$O(n^2)$$ O ( n 2 ) -time preprocessing our algorithm runs in $$O(d^*)$$ O ( d ∗ ) time.