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Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent observations indicate that hyperbolic geometry is better suited to represent hierarchical information and heterogeneous data (e.g., graphs with a scale-free degree distribution). Despite their potential for more accurate representations, hyperbolic embeddings also have downsides like being more difficult to compute and harder to use in downstream tasks. We propose embedding into a weighted space, which is closely related to hyperbolic geometry but mathematically simpler. We provide the embedding algorithm WEmbed and demonstrate, based on generated as well as over 2000 real-world graphs, that our weighted embeddings heavily outperform state-of-the-art Euclidean embeddings for heterogeneous graphs while using fewer dimensions. The running time of WEmbed and embedding quality for the remaining instances is on par with state-of-the-art Euclidean embedders.

In this paper we study the threshold model of \emph{geometric inhomogeneous random graphs} (GIRGs); a generative random graph model that is closely related to \emph{hyperbolic random graphs} (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their \emph{connectivity}, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a \emph{giant} component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability $1 - \exp(-\Omega(n^{(3-\tau)/2}))$ for graph size $n$ and a degree distribution with power-law exponent $\tau \in (2, 3)$. Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.

We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions. Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.

Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are computationally heavy and lack theoretical foundation. On the other hand, theoretical approaches do not apply to graphs with heterogeneous degree distributions, which is often the case for complex real-world networks. To address these drawbacks, we consider geometric inhomogeneous random graphs (GIRGs) as a random graph model, which captures a variety of properties observed in practice. Our first result shows that the clustering coefficient of GIRGs scales inverse exponentially with respect to the number of dimensions, when the latter is at most logarithmic in $n$. This gives a first theoretical explanation for the low dimensionality of real-world networks as observed by Almagro et al. in 2022. We further use these insights to derive a linear-time algorithm for determining the dimensionality of a given GIRG and prove that our algorithm returns the correct number of dimensions with high probability GIRG. Our algorithm bridges the gap between theory and practice, as it not only comes with a rigorous proof of correctness but also yields results comparable to that of prior empirical approaches, as indicated by our experiments on real-world instances.

A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

One of the most fundamental graph problems is finding a shortest path from a source to a target node. While in its basic forms the problem has been studied extensively and efficient algorithms are known, it becomes significantly harder as soon as parts of the graph are susceptible to failure. Although one can recompute a shortest replacement path after every outage, this is rather inefficient both in time and/or storage. One way to overcome this problem is to shift computational burden from the queries into a pre-processing step, where a data structure is computed that allows for fast querying of replacement paths, typically referred to as a Distance Sensitivity Oracle (DSO). While DSOs have been extensively studied in the theoretical computer science community, to the best of our knowledge this is the first work to construct DSOs using deep learning techniques. We show how to use deep learning to utilize a combinatorial structure of replacement paths. More specifically, we utilize the combinatorial structure of replacement paths as a concatenation of shortest paths and use deep learning to find the pivot nodes for stitching shortest paths into replacement paths., Comment: arXiv admin note: text overlap with arXiv:2007.11495 by other authors

We study computational aspects of repulsive Gibbs point processes, which are probabilistic models of interacting particles in a finite-volume region of space. We introduce an approach for reducing a Gibbs point process to the hard-core model, a well-studied discrete spin system. Given an instance of such a point process, our reduction generates a random graph drawn from a natural geometric model. We show that the partition function of a hard-core model on graphs generated by the geometric model concentrates around the partition function of the Gibbs point process. Our reduction allows us to use a broad range of algorithms developed for the hard-core model to sample from the Gibbs point process and approximate its partition function. This is, to the extend of our knowledge, the first approach that deals with pair potentials of unbounded range. We compare the resulting algorithms with recently established results and study further properties of the random geometric graphs with respect to the hard-core model.

A Voronoi diagram is a basic geometric structure that partitions the space into regions associated with a given set of sites, such that all points in a region are closer to the corresponding site than to all other sites. While being thoroughly studied in Euclidean space, they are also of interest in hyperbolic space. In fact, there are several algorithms for computing hyperbolic Voronoi diagrams that work with the various models used to describe hyperbolic geometry. However, the polar-coordinate model has not been considered before, despite its popularity in the network science community. While Voronoi diagrams have the potential to advance this field, the model is geometrically not as approachable as other models, which impedes the development of geometric algorithms. In this paper, we present an algorithm for computing Voronoi diagrams natively in the polar-coordinate model of the hyperbolic plane. The approach is based on Fortune's sweep line algorithm for Euclidean Voronoi diagrams. We characterize the hyperbolic counterparts of the concepts it utilizes and introduce adaptations necessary to account for the differences. We implemented our algorithm and compared it with the corresponding CGAL implementation. While not being as numerically stable, our method has proven to be useful as a reference, which helped resolving fundamental issues in the implementation of the state-of-the-art method.

Human lives are increasingly influenced by algorithms, which therefore need to meet higher standards not only in accuracy but also with respect to explainability. This is especially true for high-stakes areas such as real estate valuation. Unfortunately, the methods applied there often exhibit a trade-off between accuracy and explainability. One explainable approach is case-based reasoning (CBR), where each decision is supported by specific previous cases. However, such methods can be wanting in accuracy. The unexplainable machine learning approaches are often observed to provide higher accuracy but are not scrutable in their decision-making. In this paper, we apply evolutionary algorithms (EAs) to CBR predictors in order to improve their performance. In particular, we deploy EAs to the similarity functions (used in CBR to find comparable cases), which are fitted to the data set at hand. As a consequence, we achieve higher accuracy than state-of-the-art deep neural networks (DNNs), while keeping interpretability and explainability. These results stem from our empirical evaluation on a large data set of real estate offers where we compare known similarity functions, their EA-improved counterparts, and DNNs. Surprisingly, DNNs are only on par with standard CBR techniques. However, using EA-learned similarity functions does yield an improved performance.

We study algorithmic applications of a natural discretization for the hard-sphere model and the Widom-Rowlinson model in a region $\mathbb{V}\subset\mathbb{R}^d$. These models are used in statistical physics to describe mixtures of one or multiple particle types subjected to hard-core interactions. For each type, particles follow a Poisson point process with a type specific activity parameter (fugacity). The Gibbs distribution is characterized by the mixture of these point processes conditioned that no two particles are closer than a type-dependent distance threshold. A key part in better understanding the Gibbs distribution is its normalizing constant, called partition function. We give sufficient conditions that the partition function of a discrete hard-core model on a geometric graph based on a point set $X \subset \mathbb{V}$ closely approximates those of such continuous models. Previously, this was only shown for the hard-sphere model on cubic regions $\mathbb{V}=[0, \ell)^d$ when $X$ is exponential in the volume of the region $\nu(\mathbb{V})$, limiting algorithmic applications. In the same setting, our refined analysis only requires a quadratic number of points, which we argue to be tight. We use our improved discretization results to approximate the partition functions of the hard-sphere model and the Widom-Rowlinson efficiently in $\nu(\mathbb{V})$. For the hard-sphere model, we obtain the first quasi-polynomial deterministic approximation algorithm for the entire fugacity regime for which, so far, only randomized approximations are known. Furthermore, we simplify a recently introduced fully polynomial randomized approximation algorithm. Similarly, we obtain the best known deterministic and randomized approximation bounds for the Widom-Rowlinson model. Moreover, we obtain approximate sampling algorithms for the respective spin systems within the same fugacity regimes.

The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.

Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of $\sqrt{2}$. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a $(1 + o(1))$-approximation, asymptotically almost surely, and has a running time of $\mathcal{O}(m \log(n))$. The proposed algorithm is an adaptation of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.

Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent $\beta$, and high clustering that can be controlled via the temperature $T$. We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to $T = 0$. Our generators support parallelization, although this is not the focus of this paper. We note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases.

The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.

A popular paradigm for 3D point cloud registration is by extracting 3D keypoint correspondences, then estimating the registration function from the correspondences using a robust algorithm. However, many existing 3D keypoint techniques tend to produce large proportions of erroneous correspondences or outliers, which significantly increases the cost of robust estimation. An alternative approach is to directly search for the subset of correspondences that are pairwise consistent, without optimising the registration function. This gives rise to the combinatorial problem of matching with pairwise constraints. In this paper, we propose a very efficient maximum clique algorithm to solve matching with pairwise constraints. Our technique combines tree searching with efficient bounding and pruning based on graph colouring. We demonstrate that, despite the theoretical intractability, many real problem instances can be solved exactly and quickly (seconds to minutes) with our algorithm, which makes our approach an excellent alternative to standard robust techniques for 3D registration., Comment: Code and demo program are available in the supplementary material. 9 pages

A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is $\mathcal {\tilde O}(n^{2 - 1/\alpha} + n^{1/(2\alpha)} + \delta_{\max})$ with high probability, where $\alpha \in (0.5, 1)$ controls the power-law exponent of the degree distribution, and $\delta_{\max}$ is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.