Till Fluschnik

The Size Distribution, Scaling Properties and Spatial Organization of Urban Clusters: A Global and Regional Percolation Perspective

Human development has far-reaching impacts on the surface of the globe. The transformation of natural land cover occurs in different forms, and urban growth is one of the most eminent transformative processes. We analyze global land cover data and extract cities as defined by maximally connected urban clusters. The analysis of the city size distribution for all cities on the globe confirms Zipf’s law. Moreover, by investigating the percolation properties of the clustering of urban areas we assess the closeness to criticality for various countries. At the critical thresholds, the urban land cover of the countries undergoes a transition from separated clusters to a gigantic component on the country scale. We study the Zipf-exponents as a function of the closeness to percolation and find a systematic dependence, which could be the reason for deviating exponents reported in the literature. Moreover, we investigate the average size of the clusters as a function of the proximity to percolation and find country specific behavior. By relating the standard deviation and the average of cluster sizes—analogous to Taylor’s law—we suggest an alternative way to identify the percolation transition. We calculate spatial correlations of the urban land cover and find long-range correlations. Finally, by relating the areas of cities with population figures we address the global aspect of the allometry of cities, finding an exponent δ ≈ 0.85, i.e., large cities have lower densities.

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 \(O(n^4)\). 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 \(2^{O(k)} + O(n +m)\) (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 \(2^{o(k)}n^2\)-time algorithm.

Parameterized Aspects of Triangle Enumeration

Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, “subcubic” parameterized solving algorithms) as well as negative results (uselessness in terms of possibility of “faster” parameterized algorithms of certain parameters such as diameter).

Fractals for Kernelization Lower Bounds

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim. 2011], we show that, unless NP

Finding Secluded Places of Special Interest in Graphs

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Fractals for Kernelization Lower Bounds, With an Application to Length-Bounded Cut Problems

coNP / poly, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most

The Complexity of Finding Small Separators in Temporal Graphs

k

Kernelization Lower Bounds for Finding Constant-Size Subgraphs

edges such that the resulting graph has no

The Complexity of Routing with Few Collisions

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A more fine-grained complexity analysis of finding the most vital edges for undirected shortest paths

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