The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Finding secluded places of special interest in graphs

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets

Finding secluded places of special interest in graphs.

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets

Incidence of premature battery depletion in subcutaneous cardioverter-defibrillator patients: insights from a multicenter registry.

BACKGROUND The subcutaneous ICD established its role in the prevention of sudden cardiac death in recent years. The occurrence of premature battery depletion in a large subset of potentially affected devices has been a cause of concern. The incidence of premature battery depletion has not been studied systematically beyond manufacturer-reported data. METHODS Retrospective data and the most recent follow-up data on S-ICD devices from fourteen centers in Europe, the US, and Canada was studied. The incidence of generator removal or failure was reported to investigate the incidence of premature S-ICD battery depletion, defined as battery failure within 60 months or less. RESULTS Data from 1054 devices was analyzed. Premature battery depletion occurred in 3.5% of potentially affected devices over an observation period of 49 months. CONCLUSIONS The incidence of premature battery depletion of S-ICD potentially affected by a battery advisory was around 3.5% after 4 years in this study. Premature depletion occurred exclusively in devices under advisory. This is in line with the most recently published reports from the manufacturer. TRIAL REGISTRATION ClinicalTrials.gov Identifier: NCT04767516

Unveiling relationships between crime and property in England and Wales via density scale-adjusted metrics and network tools

Scale-adjusted metrics (SAMs) are a significant achievement of the urban scaling hypothesis. SAMs remove the inherent biases of per capita measures computed in the absence of isometric allometries. However, this approach is limited to urban areas, while a large portion of the world’s population still lives outside cities and rural areas dominate land use worldwide. Here, we extend the concept of SAMs to population density scale-adjusted metrics (DSAMs) to reveal relationships among different types of crime and property metrics. Our approach allows all human environments to be considered, avoids problems in the definition of urban areas, and accounts for the heterogeneity of population distributions within urban regions. By combining DSAMs, cross-correlation, and complex network analysis, we find that crime and property types have intricate and hierarchically organized relationships leading to some striking conclusions. Drugs and burglary had uncorrelated DSAMs and, to the extent property transaction values are indicators of affluence, twelve out of fourteen crime metrics showed no evidence of specifically targeting affluence. Burglary and robbery were the most connected in our network analysis and the modular structures suggest an alternative to "zero-tolerance" policies by unveiling the crime and/or property types most likely to affect each other

The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

This work studies the parameterized complexity of finding secluded solutions to classical combinatorial optimization problems on graphs such as finding minimum - separators, feedback vertex sets, dominating sets, maximum independent sets, and vertex Herein, one searches not only to minimize or maximize the size of the solution, but also to minimize the size of its neighborhood. This restriction has applications in secure routing and community detection

When can graph hyperbolicity be computed in linear time?

Hyperbolicity is a distance-based measure of how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms used in practice for computing the hyperbolicity number of an n-vertex graph have running time O(n4) . Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For example, we show that hyperbolicity can be computed in 2O(k)+O(n+m) time (where m and k denote the number of edges and the size of a vertex cover in the input graph, respectively) while at the same time, unless the Strong Exponential Time Hypothesis (SETH) fails, there is no 2o(k)⋅n2−ε -time algorithm for every ε>0

When can graph hyperbolicity be computed in linear time?

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known practical algorithms for computing the hyperbolicity number of a n-vertex graph have running time O(n4)O(n4) . Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time 2O(k)+O(n+m)2O(k)+O(n+m) (m being the number of graph edges, k being the size of a vertex cover) while at the same time, unless the SETH fails, there is no 2o(k)n22o(k)n2 -time algorithm

Non-Strict Temporal Exploration

A temporal graph G=〈G1,...,GL〉is a sequence of graphs Gi⊆G, for some given underlying graph G of order n. We consider the non-strict variant of the Temporal Exploration problem, in which we are asked to decide if G admits a sequence W of consecutively crossed edges e∈G, such that W visits all vertices at least once and that each e∈W is crossed at a timestep t′∈[L] such that t′≥t, where t is the time step during which the previous edge was crossed. This variant of the problem is shown to be NP-complete. We also consider the hardness of approximating the exploration time for yes-instances in which our order-ninput graph satisfies certain assumptions that ensure exploration schedules always exist. The first is that each pair of vertices are contained in the same component at least once in every period of nsteps, whilst the second is that the temporal diameter of our input graphis bounded by a constantc. For the latter of these two assumptions we showO(n12−ε)-inapproximability and O(n1−ε)-inapproximability in thec= 2 andc≥3 cases, respectively. For graphs with temporal diameterc= 2, we also prove an O(√nlogn) upper bound on worst-case time required for exploration, as well as anΩ(√n) lower boun