arXiv subjects

Malte Renken

Publications and source records attributed to Malte Renken.

At least 19 recordsLinked to original sources

arXiv2026

Smooth Routing in Decaying Trees

Motivated by evacuation scenarios arising in extreme events such as flooding or forest fires, we study the problem of smoothly scheduling a set of paths in graphs where connections become impassable at some point in time. A schedule is smooth if no two paths meet on an edge and the number of paths simultaneously located at a vertex does not exceed its given capacity. We study the computational complexity of the problem when the underlying graph is a tree, in particular a star or a path. We prove that already in these settings, the problem is NP-hard even with further restrictions on the capacities or on the time when all connections ceased. We provide an integer linear program (ILP) to compute the latest possible time to evacuate. Using the ILP and its relaxation, we solve sets of artificial (where each underlying graph forms either a path or star) and semi-artificial instances (where the graphs are obtained from German cities along rivers), study the runtimes, and compare the results of the ILP with those of its relaxation.

cs.DS↗

arXiv2025

The Complexity of Transitively Orienting Temporal Graphs

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 $t$''. 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. An orientation of a temporal graph is called temporally transitive if, whenever $u$ has a directed edge towards $v$ with time-label $t_1$ and $v$ has a directed edge towards $w$ with time-label $t_2\geq t_1$, then $u$ also has a directed edge towards $w$ with some time-label $t_3\geq t_2$. If we just demand that this implication holds whenever $t_2 > t_1$, we call the orientation strictly temporally transitive, as it is based on the 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 $\mathcal{G}$ is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether $\mathcal{G}$ 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.

cs.DS↗

arXiv2024

Locally Rainbow Paths

We introduce the algorithmic problem of finding a locally rainbow path of length $\ell$ 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 least $r$ 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.

cs.DS↗

arXiv2023

Sharp Thresholds in Random Simple Temporal Graphs

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 $G~G_{n,p}$ by considering a random permutation $π$ of the edges and interpreting the ranks in $π$ as presence times. 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 significant because temporal graphs do not admit spanners of size $O(n)$ in general (Kempe et al, STOC 2000). In fact, they do not even admit spanners of size $o(n^2)$ (Axiotis et al, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, all non-negligible obstructions, must be statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners 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 $p = 4\log n/n$.

cs.DM↗

arXiv2023

Giant Components in Random Temporal Graphs

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 $p = \log n/n$ the size of the largest temporally connected component increases from $o(n)$ to~$n-o(n)$. This threshold holds for both open and closed connected components, i.e. components that allow, respectively forbid, their connecting paths to use external nodes.

cs.DM↗

arXiv2023

Two Influence Maximization Games on Graphs Made Temporal

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 2-player competitive diffusion and Voronoi games on different temporal graph classes. Even when restricting our studies to temporal trees 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 trees and cycles. Our two main technical results are (algorithmic) proofs for the existence of Nash equilibria in 2-player competitive diffusion and temporal Voronoi games when the edges are restricted not to disappear over time.

cs.GT↗

arXiv2022

Terrain-like Graphs and the Median Genocchi Numbers

A graph with vertex set $\{1,\ldots,n\}$ is terrain-like if, for any edge pair $\{a,c\},\{b,d\}$ with $a<b<c<d$, the edge $\{a,d\}$ also exists. Terrain-like graphs frequently appear in geometry in the context of visibility graphs. We show that terrain-like graphs are counted by the median Genocchi numbers. To this end, we prove a bijection between terrain-like graphs and Dumont derangements of the second kind.

math.CO↗

arXiv2022

Using a geometric lens to find k disjoint shortest paths

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.

math.CO↗

arXiv2022

Delay-Robust Routes in Temporal Graphs

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.

cs.DS↗

arXiv2022

Temporal Connectivity: Coping with Foreseen and Unforeseen Delays

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.

cs.DS↗

arXiv2021

Feedback Edge Sets in Temporal Graphs

The classical, linear-time solvable Feedback Edge Set problem is concerned with finding a minimum number of edges intersecting all cycles in a (static, unweighted) graph. We provide a first study of this problem in the setting of temporal graphs, where edges are present only at certain points in time. We find that there are four natural generalizations of Feedback Edge Set, all of which turn out to be NP-hard. We also study the tractability of these problems with respect to several parameters (solution size, lifetime, and number of graph vertices, among others) and obtain some parameterized hardness but also fixed-parameter tractability results.

cs.DM↗

arXiv2021

Temporal Reachability Minimization: Delaying vs. Deleting

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.

cs.DS↗

arXiv2021

Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives

Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We now study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the proximity graph classes relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the problems remain NP-hard on these graphs, except for 3-Colorability and Hamiltonian Cycle on relatively closest graphs, where the former is trivial and the latter is left open. Moreover, for every NP-hard case we additionally show that no $2^{o(n^{1/4})}$-time algorithm exists unless the ETH fails, where n denotes the number of vertices.

cs.CC↗

arXiv2021

On Finding Separators in Temporal Split and Permutation Graphs

Removing all connections between two vertices s and z in a graph by removing a minimum number of vertices is a fundamental problem in algorithmic graph theory. This (s,z)-separation problem is well-known to be polynomial solvable and serves as an important primitive in many applications related to network connectivity. We study the NP-hard temporal (s,z)-separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle this problem by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general the problem remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of "change over time".

cs.DM↗

arXiv2021

Parameterized Algorithms for Diverse Multistage Problems

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.

cs.DS↗

arXiv2020

Persistent Graphs and Cyclic Polytope Triangulations

We prove a bijection between the triangulations of the 3-dimensional cyclic polytope C(n+2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We additionally give an algorithm to efficiently enumerate all persistent graphs on n vertices and thus all triangulations of C(n+2, 3).

cs.DM↗

arXiv2020

Temporal Graph Classes: A View Through Temporal Separators

We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph---there we observe polynomial-time solvability in the case of bounded treewidth---as well as restrictions concerning the "temporal evolution" along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases.

cs.CC↗

arXiv2020

Advancing Through Terrains

We study terrain visibility graphs, a well-known graph class closely related to polygon visibility graphs in computational geometry, for which a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted attention in the context of time series analysis with various practical applications in areas such as physics, geography and medical sciences. We make progress in understanding terrain visibility graphs by providing several graph-theoretic results. For example, we show that they cannot contain antiholes of size larger than five. Moreover, we obtain two algorithmic results. We devise a fast output-sensitive shortest path algorithm on terrain-like graphs and a polynomial-time algorithm for \textsc{Dominating Set} on special terrain visibility graphs (called funnel visibility graphs).

cs.CG↗