Two Moves per Time Step Make a Difference

Erlebach, Thomas; Kammer, Frank; Luo, Kelin; Sajenko, Andrej; Spooner, Jakob T.

doi:10.4230/lipics.icalp.2019.141

Cited by 7 publications

(7 citation statements)

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“…Third, we believe that our geometric perspective presented in Section 3.1 can be applied to other temporal graph problems. In particular, for temporal graph problems which ask for specific temporal paths, e.g., temporal paths that obey certain robustness properties [26], or temporal paths that visit all vertices at least once [21,22,36] parameterized by the temporal diameter, that is, the length of the longest shortest temporal path between two arbitrary vertices.…”

Section: Discussionmentioning

confidence: 99%

“…This problem remains computationally hard even if the underlying graph is a star [3,12]. If the underlying graph is connected at each time step and the walk can only contain one edge in each time step, then a fast exploration is guaranteed [21,23,22]. However, on these so-called always-connected temporal graphs, the decision problem remains NP-hard, even if the underlying graph has pathwidth two [10].…”

Section: Short Restless Temporal Pathmentioning

confidence: 99%

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Restless Temporal Path Parameterized Above Lower Bounds

Zschoche¹

2022

Preprint

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Reachability questions are one of the most fundamental algorithmic primitives in temporal graphsgraphs whose edge set changes over discrete time steps. A core problem here is the NP-hard Short Restless Temporal Path: given a temporal graph G, two distinct vertices s and z, and two numbers δ and k, is there a δ-restless temporal s-z path of length at most k? A temporal path is a path whose edges appear in chronological order and a temporal path is δ-restless if two consecutive path edges appear at most δ time steps apart from each other. Among others, this problem has applications in neuroscience and epidemiology. While Short Restless Temporal Path is known to be computationally hard, e.g., it is NP-hard for only three time steps and W[1]-hard when parameterized by the feedback vertex number of the underlying graph, it is fixed-parameter tractable when parameterized by the path length k. We improve on this by showing that Short Restless Temporal Path can be solved in (randomized) 4 k−d |G| O(1) time, where d is the minimum length of a temporal s-z path.

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Section: Discussionmentioning

confidence: 99%

Section: Short Restless Temporal Pathmentioning

confidence: 99%

Restless Temporal Path Parameterized Above Lower Bounds

Zschoche¹

2022

Preprint

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“…Akrida et al [1] consider Return-To-Base TEXP in which a candidate solution must return to the vertex from which it initially departed. Erlebach et al [7] prove an O(dn 1.75 ) bound on the number of time steps required to explore any temporal graph with degree bounded by d in each step, a considerable improvement over the previously best known O( n 2 log d log n ) bound [8]. In [9], a non-strict variant of TEXP is studied-here, a computed walk may make an unlimited number of edge traversals in each given time step.…”

Section: Related Workmentioning

confidence: 99%

“…The graph exploration problem in the context of temporal graphs (i.e. graphs whose edge set can change over time) has also received significant attention in recent years [1,2,[6][7][8][9]21]. This problem, known as Temporal Exploration (TEXP), but restricted to k-edgedeficient temporal graphs (which we define formally later) is the focus of this paper.…”

Section: Introductionmentioning

confidence: 99%

Exploration of k-edge-deficient temporal graphs

Erlebach

Spooner

2022

Acta Informatica

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A temporal graph with lifetime L is a sequence of L graphs $$G_1, \ldots ,G_L$$ G 1 , … , G L , called layers, all of which have the same vertex set V but can have different edge sets. The underlying graph is the graph with vertex set V that contains all the edges that appear in at least one layer. The temporal graph is always connected if each layer is a connected graph, and it is k-edge-deficient if each layer contains all except at most k edges of the underlying graph. For a given start vertex s, a temporal exploration is a temporal walk that starts at s, traverses at most one edge in each layer, and visits all vertices of the temporal graph. We show that always-connected, k-edge-deficient temporal graphs with sufficient lifetime can always be explored in $$O(kn \log n)$$ O ( k n log n ) time steps. We also construct always-connected, k-edge-deficient temporal graphs for which any exploration requires $$\varOmega (n \log k)$$ Ω ( n log k ) time steps. For always-connected, 1-edge-deficient temporal graphs, we show that O(n) time steps suffice for temporal exploration.

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“…This requirement makes temporal reachability non-symmetric and non-transitive, which stands in contrast to reachability in normal (static) graphs. Reachability is arguably one of the most central concepts in temporal graph algorithmics and has been studied under various aspects, such as path finding [5,9,12,40], vertex separation [28,34,41], finding spanning subgraphs [4,11], temporal graph exploration [2,7,23,24,25,26], and others [3,10,32,35].…”

Section: Introductionmentioning

confidence: 99%

Temporal Reachability Minimization: Delaying vs. Deleting

Molter,

Renken,

Zschoche

2021

Preprint

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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. ACM Subject ClassificationMathematics of computing → Graph algorithms; Mathematics of computing → Paths and connectivity problems; Mathematics of computing → Network flows

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Two Moves per Time Step Make a Difference

Cited by 7 publications

References 0 publications

Restless Temporal Path Parameterized Above Lower Bounds

Restless Temporal Path Parameterized Above Lower Bounds

Exploration of k-edge-deficient temporal graphs

Temporal Reachability Minimization: Delaying vs. Deleting

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