We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system $(V,\mathcal{F},\mathcal{B})$ with two families $\mathcal{F},\mathcal{B}$ of subsets of $V$. The task is to find a hitting set for $\mathcal{F}$ that minimizes the maximum number of elements in any of the sets of $\mathcal{B}$. Our focus is on determining the complexity of some special cases of Sparse-HS with respect to the sparseness $k$, which is the optimum number of hitting set elements in any set of $\mathcal{B}$. For the Sparse Vertex Cover (Sparse-VC) problem, $V$ is given by the vertex set of a graph, and $\mathcal{F}$ is its edge set. We prove NP-hardness for sparseness $k\geq 2$ and polynomial time solvability for $k=1$. We also provide a polynomial-time $2$-approximation for any $k$. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family $\mathcal{B}$ is given by vertex neighbourhoods. For this problem we prove NP-hardness for constant $k$ and provide a polynomial-time $(2-\frac{1}{k})$-approximation. This is better than any approximation possible for Sparse-VC or Vertex Cover (under UGC). We then consider two problems derived from Sparse-HS related to the highway dimension, a graph parameter modelling transportation networks. Most algorithms for graphs of low highway dimension compute solutions to the $r$-Shortest Path Cover ($r$-SPC) problem, where $r>0$, $\mathcal{F}$ contains all shortest paths of length between $r$ and $2r$, and $\mathcal{B}$ contains all balls of radius $2r$. There is an XP algorithm that computes solutions to $r$-SPC of sparseness at most $h$ if the input graph has highway dimension $h$, but the existence if an FPT algorithm was open. We prove that $r$-SPC and also the related $r$-Highway Dimension ($r$-HD) problem are both W[1]-hard. Furthermore, we prove that $r$-SPC admits a polynomial-time $O(\log n)$-approximation.

Resource demands are crucial parameters for modeling and predicting the performance of software systems. Currently, resource demand estimators are usually executed once for system analysis. However, the monitored system, as well as the resource demand itself, are subject to constant change in runtime environments. These changes additionally impact the applicability, the required parametrization as well as the resulting accuracy of individual estimation approaches. Over time, this leads to invalid or outdated estimates, which in turn negatively influence the decision-making of adaptive systems. In this article, we present SARDE , a framework for self-adaptive resource demand estimation in continuous environments. SARDE dynamically and continuously tunes, selects, and executes an ensemble of resource demand estimation approaches to adapt to changes in the environment. This creates an autonomous and unsupervised ensemble estimation technique, providing reliable resource demand estimations in dynamic environments. We evaluate SARDE using two realistic datasets. One set of different micro-benchmarks reflecting different possible system states and one dataset consisting of a continuously running application in a changing environment. Our results show that by continuously applying online optimization, selection and estimation, SARDE is able to efficiently adapt to the online trace and reduce the model error using the resulting ensemble technique.

Abstract We study the k -Center problem, where the input is a graph $$G=(V,E)$$ G = ( V , E ) with positive edge weights and an integer k , and the goal is to select k center vertices $$C \subseteq V$$ C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is $$\mathsf {NP}$$ NP -hard and cannot be approximated within a factor less than 2. Typical applications of the k -Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k -Center is $$\mathsf {W[1]}$$ W [ 1 ] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k , the highway dimension $$hd$$ hd and the pathwidth $$pw$$ pw (Feldmann and Marx 2020). We extend their result and show that even if we additionally parameterize by the skeleton dimension $$\kappa $$ κ , the k -Center problem remains $$\mathsf {W[1]}$$ W [ 1 ] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k -Center that has runtime $$f(k,hd,pw,\kappa ) \cdot \vert V \vert ^{o(pw+ \kappa + \sqrt{k+hd})}$$ f ( k , h d , p w , κ ) · | V | o ( p w + κ + k + h d ) for any computable function f .

Contraction hierarchies (CH) is a prominent preprocessing-based technique that accelerates the computation of shortest paths in road networks by reducing the search space size of a bidirectional Dijkstra run. To explain the practical success of CH, several theoretical upper bounds for the maximum search space size were derived in previous work. For example, it was shown that in minor-closed graph families search space sizes in 𝒪(√n) can be achieved (with n denoting the number of nodes in the graph), and search space sizes in 𝒪(h log D) in graphs of highway dimension h and diameter D. In this paper, we primarily focus on lower bounds. We prove that the average search space size in a so called weak CH is in Ω(b_α) for α ≥ 2/3 where b_α is the size of a smallest α-balanced node separator. This discovery allows us to describe the first approximation algorithm for the average search space size. Our new lower bound also shows that the 𝒪(√n) bound for minor-closed graph families is tight. Furthermore, we deeper investigate the relationship of CH and the highway dimension and skeleton dimension of the graph, and prove new lower bound and incomparability results. Finally, we discuss how lower bounds for strong CH can be obtained from solving a HittingSet problem defined on a set of carefully chosen subgraphs of the input network.

The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs $G=(V,E)$ of skeleton dimension $\mathcal{O}(\log^2 \vert V \vert)$ it is NP-hard to approximate the $k$-Center problem within a factor less than $2$.

Shortest path planning is a fundamental building block in many applications. Hence developing efficient methods for computing shortest paths in e.g. road or grid networks is an important challenge. The most successful techniques for fast query answering rely on preprocessing. But for many of these techniques it is not fully understood why they perform so remarkably well and theoretical justification for the empirical results is missing. An attempt to explain the excellent practical performance of preprocessing based techniques on road networks (as transit nodes, hub labels, or contraction hierarchies) in a sound theoretical way are parametrized analyses, e.g., considering the highway dimension or skeleton dimension of a graph. But these parameters tend to be large (order of Θ(√n)) when the network contains grid-like substructures — which inarguably is the case for real-world road networks around the globe. In this paper, we use the very intuitive notion of bounded growth graphs to describe road networks and also grid graphs. We show that this model suffices to prove sublinear search spaces for the three above mentioned state-of-the-art shortest path planning techniques. For graphs with a large highway or skeleton dimension, our results turn out to be superior. Furthermore, our preprocessing methods are close to the ones used in practice and only require randomized polynomial time.