arXiv subjects

Petra Mutzel

Publications and source records attributed to Petra Mutzel.

At least 37 records · Page 2Linked to original sources

arXiv2021

Temporal Graph Kernels for Classifying Dissemination Processes

Many real-world graphs or networks are temporal, e.g., in a social network persons only interact at specific points in time. This information directs dissemination processes on the network, such as the spread of rumors, fake news, or diseases. However, the current state-of-the-art methods for supervised graph classification are designed mainly for static graphs and may not be able to capture temporal information. Hence, they are not powerful enough to distinguish between graphs modeling different dissemination processes. To address this, we introduce a framework to lift standard graph kernels to the temporal domain. Specifically, we explore three different approaches and investigate the trade-offs between loss of temporal information and efficiency. Moreover, to handle large-scale graphs, we propose stochastic variants of our kernels with provable approximation guarantees. We evaluate our methods on a wide range of real-world social networks. Our methods beat static kernels by a large margin in terms of accuracy while still being scalable to large graphs and data sets. Hence, we confirm that taking temporal information into account is crucial for the successful classification of dissemination processes.

cs.SI↗

arXiv2020

Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs

We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with $n$ nodes and $k$ edge crossings in time $O(3^k \cdot n^{3/2} \log n)$.

cs.DS↗

arXiv2020

Weisfeiler and Leman go sparse: Towards scalable higher-order graph embeddings

Graph kernels based on the $1$-dimensional Weisfeiler-Leman algorithm and corresponding neural architectures recently emerged as powerful tools for (supervised) learning with graphs. However, due to the purely local nature of the algorithms, they might miss essential patterns in the given data and can only handle binary relations. The $k$-dimensional Weisfeiler-Leman algorithm addresses this by considering $k$-tuples, defined over the set of vertices, and defines a suitable notion of adjacency between these vertex tuples. Hence, it accounts for the higher-order interactions between vertices. However, it does not scale and may suffer from overfitting when used in a machine learning setting. Hence, it remains an important open problem to design WL-based graph learning methods that are simultaneously expressive, scalable, and non-overfitting. Here, we propose local variants and corresponding neural architectures, which consider a subset of the original neighborhood, making them more scalable, and less prone to overfitting. The expressive power of (one of) our algorithms is strictly higher than the original algorithm, in terms of ability to distinguish non-isomorphic graphs. Our experimental study confirms that the local algorithms, both kernel and neural architectures, lead to vastly reduced computation times, and prevent overfitting. The kernel version establishes a new state-of-the-art for graph classification on a wide range of benchmark datasets, while the neural version shows promising performance on large-scale molecular regression tasks.

cs.DS↗

arXiv2020

Spatio-Temporal Top-k Similarity Search for Trajectories in Graphs

We study the problem of finding the $k$ most similar trajectories to a given query trajectory. Our work is inspired by the work of Grossi et al. [6] that considers trajectories as walks in a graph. Each visited vertex is accompanied by a time-interval. Grossi et al. define a similarity function that captures temporal and spatial aspects. We improve this similarity function to derive a new spatio-temporal distance function for which we can show that a specific type of triangle inequality is satisfied. This distance function is the basis for our index structures, which can be constructed efficiently, need only linear memory, and can quickly answer queries for the top-$k$ most similar trajectories. Our evaluation on real-world and synthetic data sets shows that our algorithms outperform the baselines with respect to indexing time by several orders of magnitude while achieving similar or better query time and quality of results.

cs.DS↗

arXiv2020

Maximum Cut Parameterized by Crossing Number

Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.

cs.DS↗

arXiv2020

TUDataset: A collection of benchmark datasets for learning with graphs

Recently, there has been an increasing interest in (supervised) learning with graph data, especially using graph neural networks. However, the development of meaningful benchmark datasets and standardized evaluation procedures is lagging, consequently hindering advancements in this area. To address this, we introduce the TUDataset for graph classification and regression. The collection consists of over 120 datasets of varying sizes from a wide range of applications. We provide Python-based data loaders, kernel and graph neural network baseline implementations, and evaluation tools. Here, we give an overview of the datasets, standardized evaluation procedures, and provide baseline experiments. All datasets are available at www.graphlearning.io. The experiments are fully reproducible from the code available at www.github.com/chrsmrrs/tudataset.

cs.LG↗

arXiv2020

On the Enumeration and Counting of Bicriteria Temporal Paths

We discuss the complexity of path enumeration and counting in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some real cost. We introduce two bicriteria temporal min-cost path problems in which we are interested in the set of all efficient paths with low cost and short duration or early arrival time, respectively. However, the number of efficient paths can be exponential in the size of the input. For the case of strictly positive edge costs we are able to provide algorithms that enumerate the set of efficient paths with polynomial time delay and polynomial space. If we are only interested in the set of Pareto-optimal solutions and not in the paths themselves, then these can be determined in polynomial time if all edge costs are nonnegative. In addition, for each Pareto-optimal solution, we are able to find an efficient path in polynomial time. On the negative side, we prove that counting the number of efficient paths is #P-complete, even in the non-weighted single criterion case.

cs.CC↗

arXiv2019

A Unifying View of Explicit and Implicit Feature Maps of Graph Kernels

Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. We investigate how convolution kernels for structured data are composed from base kernels and construct corresponding feature maps. On this basis we propose exact and approximative feature maps for widely used graph kernels based on the kernel trick. We analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and graph invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time. Moreover, we propose and analyze algorithms for computing random walk, shortest-path and subgraph matching kernels by explicit and implicit feature maps. Our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively.

cs.LG↗

arXiv2019

Performance of a Quantum Annealer for Ising Ground State Computations on Chimera Graphs

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue.

cs.DS↗

arXiv2018

A note on block-and-bridge preserving maximum common subgraph algorithms for outerplanar graphs

Schietgat, Ramon and Bruynooghe proposed a polynomial-time algorithm for computing a maximum common subgraph under the block-and-bridge preserving subgraph isomorphism (BBP-MCS) for outerplanar graphs. We show that the article contains the following errors: (i) The running time of the presented approach is claimed to be $\mathcal{O}(n^{2.5})$ for two graphs of order $n$. We show that the algorithm of the authors allows no better bound than $\mathcal{O}(n^4)$ when using state-of-the-art general purpose methods to solve the matching instances arising as subproblems. This is even true for the special case, where both input graphs are trees. (ii) The article suggests that the dissimilarity measure derived from BBP-MCS is a metric. We show that the triangle inequality is not always satisfied and, hence, it is not a metric. Therefore, the dissimilarity measure should not be used in combination with techniques that rely on or exploit the triangle inequality in any way. Where possible, we give hints on techniques that are suitable to improve the algorithm.

cs.DS↗

arXiv2018

A new Integer Linear Program for the Steiner Tree Problem with Revenues, Budget and Hop Constraints

The Steiner tree problem with revenues, budgets and hop constraints (STPRBH) is a variant of the classical Steiner tree problem. This problem asks for a subtree in a given graph with maximum revenues corresponding to its nodes, where its total edge costs respect the given budget, and the number of edges between each node and its root does not exceed the hop limit. We introduce a new binary linear program with polynomial size based on partial-ordering, which (up to our knowledge) for the first time solves all STPRBH instances from the DIMACS benchmark set to optimality. The set contains graphs with up to 500 nodes and 12500 edges.

cs.DM↗

arXiv2018

A Flow Formulation for Horizontal Coordinate Assignment with Prescribed Width

We consider the coordinate assignment phase of the well known Sugiyama framework for drawing directed graphs in a hierarchical style. The extensive literature in this area has given comparatively little attention to a prescribed width of the drawing. We present a minimum cost flow formulation that supports prescribed width and optionally other criteria like lower and upper bounds on the distance of neighbouring nodes in a layer or enforced vertical edges segments. In our experiments we demonstrate that our approach can compete with state-of-the-art algorithms.

cs.DS↗

arXiv2018

Bishellable drawings of $K_n$

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-1}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-2}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-3}{\mathstrut 2}\right \rfloor$. Ábrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s\geq5$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellable, but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.

math.CO↗

arXiv2018

The Crossing Number of Semi-Pair-Shellable Drawings of Complete Graphs

The Harary-Hill Conjecture states that for $n\geq 3$ every drawing of $K_n$ has at least \begin{align*} H(n) := \frac{1}{4}\Big\lfloor\frac{n}{2}\Big\rfloor\Big\lfloor\frac{n-1}{2}\Big\rfloor\Big\lfloor\frac{n-2}{2}\Big\rfloor\Big\lfloor\frac{n-3}{2}\Big\rfloor \end{align*} crossings. In general the problem remains unsolved, however there has been some success in proving the conjecture for restricted classes of drawings. The most recent and most general of these classes is seq-shellability. In this work, we improve these results and introduce the new class of semi-pair-shellable drawings. We prove the Harary-Hill Conjecture for this new class using novel results on $k$-edges. So far, approaches for proving the Harary-Hill Conjecture for specific classes rely on a fixed reference face. We successfully apply new techniques in order to loosen this restriction, which enables us to select different reference faces when considering subdrawings. Furthermore, we introduce the notion of $k$-deviations as the difference between an optimal and the actual number of $k$-edges. Using $k$-deviations, we gain interesting insights into the essence of $k$-edges, and we further relax the necessity of fixed reference faces.

cs.CG↗

arXiv2018

A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

We propose a fixed-parameter tractable algorithm for the \textsc{Max-Cut} problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithm recursively reduces a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. The \textsc{Max-Cut} problem is then solved on the planar graphs using established polynomial-time algorithms. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithm computes a maximum cut in an embedded 1-planar graph with $n$ nodes and $k$ edge crossings in time $\mathcal{O}(3^k \cdot n^{3/2} \log n)$.

cs.DS↗

arXiv2018

Largest Weight Common Subtree Embeddings with Distance Penalties

The largest common embeddable subtree problem asks for the largest possible tree embeddable into two input trees and generalizes the classical maximum common subtree problem. Several variants of the problem in labeled and unlabeled rooted trees have been studied, e.g., for the comparison of evolutionary trees. We consider a generalization, where the sought embedding is maximal with regard to a weight function on pairs of labels. We support rooted and unrooted trees with vertex and edge labels as well as distance penalties for skipping vertices. This variant is important for many applications such as the comparison of chemical structures and evolutionary trees. Our algorithm computes the solution from a series of bipartite matching instances, which are solved efficiently by exploiting their structural relation and imbalance. Our analysis shows that our approach improves or matches the running time of the formally best algorithms for several problem variants. Specifically, we obtain a running time of $\mathcal O(|T|\,|T'|Δ)$ for two rooted or unrooted trees $T$ and $T'$, where $Δ=\min\{Δ(T),Δ(T')\}$ with $Δ(X)$ the maximum degree of $X$. If the weights are integral and at most $C$, we obtain a running time of $\mathcal O(|T|\,|T'|\sqrtΔ\log (C\min\{|T|,|T'|\}))$ for rooted trees.

cs.DS↗

arXiv2018

The Crossing Number of Seq-Shellable Drawings of Complete Graphs

The Harary-Hill conjecture states that for every $n>0$ the complete graph on $n$ vertices $K_n$, the minimum number of crossings over all its possible drawings equals \begin{align*} H(n) := \frac{1}{4}\Big\lfloor\frac{n}{2}\Big\rfloor\Big\lfloor\frac{n-1}{2}\Big\rfloor\Big\lfloor\frac{n-2}{2}\Big\rfloor\Big\lfloor\frac{n-3}{2}\Big\rfloor\text{.} \end{align*} So far, the lower bound of the conjecture could only be verified for arbitrary drawings of $K_n$ with $n\leq 12$. In recent years, progress has been made in verifying the conjecture for certain classes of drawings, for example $2$-page-book, $x$-monotone, $x$-bounded, shellable and bishellable drawings. Up to now, the class of bishellable drawings was the broadest class for which the Harary-Hill conjecture has been verified, as it contains all beforehand mentioned classes. In this work, we introduce the class of seq-shellable drawings and verify the Harary-Hill conjecture for this new class. We show that bishellability implies seq-shellability and exhibit a non-bishellable but seq-shellable drawing of $K_{11}$, therefore the class of seq-shellable drawings strictly contains the class of bishellable drawings.

cs.CG↗

arXiv2018

Recognizing Cuneiform Signs Using Graph Based Methods

The cuneiform script constitutes one of the earliest systems of writing and is realized by wedge-shaped marks on clay tablets. A tremendous number of cuneiform tablets have already been discovered and are incrementally digitalized and made available to automated processing. As reading cuneiform script is still a manual task, we address the real-world application of recognizing cuneiform signs by two graph based methods with complementary runtime characteristics. We present a graph model for cuneiform signs together with a tailored distance measure based on the concept of the graph edit distance. We propose efficient heuristics for its computation and demonstrate its effectiveness in classification tasks experimentally. To this end, the distance measure is used to implement a nearest neighbor classifier leading to a high computational cost for the prediction phase with increasing training set size. In order to overcome this issue, we propose to use CNNs adapted to graphs as an alternative approach shifting the computational cost to the training phase. We demonstrate the practicability of both approaches in an extensive experimental comparison regarding runtime and prediction accuracy. Although currently available annotated real-world data is still limited, we obtain a high accuracy using CNNs, in particular, when the training set is enriched by augmented examples.

cs.CV↗