Searcharxiv⌕ Search

arXiv subjects

Petra Mutzel

Publications and source records attributed to Petra Mutzel.

At least 19 recordsLinked to original sources
arXiv2026

A Separator-based Algorithm for the Graph Edit Distance Problem

The Graph Edit Distance (GED) is a widely used graph similarity measure asking for the minimum cost of a sequence of edits transforming one (labeled) graph into another. The considered edit operations are deletion, insertion, and relabeling of nodes and edges. Special cases include the Graph Isomorphism problem, as well as many other graph problems that ask for the existence or minimum cost of a certain substructure, like the Traveling Salesman or Maximum Clique problem. We present a novel exponential time algorithm to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators. In particular, the claimed runtime holds if one of the graphs is $K_h$-minor free (e.g., planar), or has bounded treewidth, which is the case for many real-world applications (e.g., all instances in GEDLIB). This substantially improves the best known worst-case running time bounds of $O^*(n!)$ for these graph classes.

cs.DS↗
arXiv2026

Strong ILP Formulations for the p-Regions Problem

Regionalization is a fundamental task in spatial analysis that seeks to partition a larger area - such as a country - into smaller regions that are homogeneous with respect to a given attribute. A popular model for regionalization is the p-regions problem, in which regions are formed by grouping the areas of an input planar subdivision. Given the subdivision's adjacency graph G and pairwise dissimilarities between vertices, the goal is to partition G into a fixed number p of connected subgraphs, such as to minimize the sum of dissimilarities over all vertex pairs in the same subgraph. The problem is NP-hard and even small instances are difficult to solve to provable optimality. In this paper, we present the new ILP model ER-S for the p-regions problem, exploiting a connection between the p-regions objective and the k-partitioning problem. Furthermore, we strengthen the known ILP model Tree with a new type of subtour elimination inequality specific to the p-regions problem. Combining ER-S and the strengthened version of Tree yields the model ER-S-Tree, which dominates the state-of-the-art models in polyhedral strength. This theoretical advantage is reflected in its superior performance in our experimental evaluation. In particular, the new models ER-S and ER-S-Tree enable the solution of problem instances for major European countries that were previously intractable.

cs.DM↗
arXiv2026

Optimality-Preserving Data Reduction for Maximum k-Cut (Full Version)

Preprocessing has become an increasingly important part of solving Maximum Cut to optimality, enabling exact solvers to tackle significantly larger instances. This suggests that exact solvers for the more general Maximum k-Cut problem could also benefit from sophisticated preprocessing. However, to the best of our knowledge, no preprocessing techniques that are effective for k > 2 have been published. In this paper, we introduce structured cut sets, a novel data reduction technique for Maximum k-Cut. We provide criteria under which deleting cut sets is optimality-preserving, yielding a decomposition into connected components that can be solved independently and whose solutions can be combined into an optimal solution for the original graph. Furthermore, we extend several preprocessing techniques from Maximum Cut to Maximum k-Cut. To show that our rules are optimality-preserving, we develop a new proof framework based on the addition of weighted graphs. We complement our theoretical results by engineering a preprocessing framework for Maximum k-Cut and show its effectiveness in a computational study. The preprocessed instances are typically significantly smaller. Integrating our preprocessing into an exact solver yields significant speed-ups and enables solving more instances to optimality.

cs.DS↗
arXiv2025

Accelerating Graph Similarity Search through Integer Linear Programming

The Graph Edit Distance (GED) is an important metric for measuring the similarity between two (labeled) graphs. It is defined as the minimum cost required to convert one graph into another through a series of (elementary) edit operations. Its effectiveness in assessing the similarity of large graphs is limited by the complexity of its exact calculation, which is NP-hard theoretically and computationally challenging in practice. The latter can be mitigated by switching to the Graph Similarity Search under GED constraints, which determines whether the edit distance between two graphs is below a given threshold. A popular framework for solving Graph Similarity Search under GED constraints in a graph database for a query graph is the filter-and-verification framework. Filtering discards unpromising graphs, while the verification step certifies the similarity between the filtered graphs and the query graph. To improve the filtering step, we define a lower bound based on an integer linear programming formulation. We prove that this lower bound dominates the effective branch match-based lower bound and can also be computed efficiently. Consequently, we propose a graph similarity search algorithm that uses a hierarchy of lower bound algorithms and solves a novel integer programming formulation that exploits the threshold parameter. An extensive computational experience on a well-assessed test bed shows that our approach significantly outperforms the state-of-the-art algorithm on most of the examined thresholds.

cs.DB↗
arXiv2025

Bicriteria Polygon Aggregation with Arbitrary Shapes

We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime $\mathcal{\tilde{O}}(n^4)$, where $n$ is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that $\mathcal{O}(n^2)$ combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.

cs.CG↗
arXiv2025

Weisfeiler and Leman Follow the Arrow of Time: Expressive Power of Message Passing in Temporal Event Graphs

An important characteristic of temporal graphs is how the directed arrow of time influences their causal topology, i.e., which nodes can possibly influence each other causally via time-respecting paths. The resulting patterns are often neglected by temporal graph neural networks (TGNNs). To formally analyze the expressive power of TGNNs, we lack a generalization of graph isomorphism to temporal graphs that fully captures their causal topology. Addressing this gap, we introduce the notion of consistent event graph isomorphism, which utilizes a time-unfolded representation of time-respecting paths in temporal graphs. We compare this definition with existing notions of temporal graph isomorphisms. We illustrate and highlight the advantages of our approach and develop a temporal generalization of the Weisfeiler-Leman algorithm to heuristically distinguish non-isomorphic temporal graphs. Building on this theoretical foundation, we derive a novel message passing scheme for temporal graph neural networks that operates on the event graph representation of temporal graphs. An experimental evaluation shows that our approach performs well in a temporal graph classification experiment.

cs.LG↗
arXiv2025

A Customized SAT-based Solver for Graph Coloring

We introduce ZykovColor, a novel SAT-based algorithm to solve the graph coloring problem working on top of an encoding that mimics the Zykov tree. Our method is based on an approach of H\'ebrard and Katsirelos (2020) that employs a propagator to enforce transitivity constraints, incorporate lower bounds for search tree pruning, and enable inferred propagations. We leverage the recently introduced IPASIR-UP interface for CaDiCaL to implement these techniques with a SAT solver. Furthermore, we propose new features that take advantage of the underlying SAT solver. These include modifying the integrated decision strategy with vertex domination hints and using incremental bottom-up search that allows to reuse learned clauses from previous calls. Additionally, we integrate a more effective clique computation and an algorithm for computing the fractional chromatic number to improve the lower bounds used for pruning during the search. We validate the effectiveness of each new feature through an experimental analysis. ZykovColor outperforms other state-of-the-art graph coloring implementations on the DIMACS benchmark set. Further experiments on random Erd\H{o}s-R\'enyi graphs show that our new approach matches or outperforms state-of-the-art SAT-based methods for both very sparse and highly dense graphs. We give an additional configuration of ZykovColor that dominates other SAT-based methods on the Erd\H{o}s-R\'enyi graphs.

cs.DM↗
arXiv2024

A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order

We present parametric breadth-first search (PBFS), a new algorithm for solving the parametric minimum cut problem in a network with source-sink-monotone capacities. The objective is to find the set of breakpoints, i.e., the points at which the minimum cut changes. It is well known that this problem can be solved in the same asymptotic runtime as the static minimum cut problem. However, existing algorithms that achieve this runtime bound involve fairly complicated steps that are inefficient in practice. PBFS uses a simpler approach that discovers the breakpoints in ascending order, which allows it to achieve the desired runtime bound while still performing well in practice. We evaluate our algorithm on benchmark instances from polygon aggregation and computer vision. Polygon aggregation was recently proposed as an application for parametric minimum cut, but the monotonicity property has not been exploited fully. PBFS outperforms the state of the art on most benchmark instances, usually by a factor of 2-3. It is particularly strong on instances with many breakpoints, which is the case for polygon aggregation. Compared to the existing min-cut-based approach for polygon aggregation, PBFS scales much better with the instance size. On large instances with millions of vertices, it is able to compute all breakpoints in a matter of seconds.

cs.DS↗
arXiv2024

Revisiting ILP Models for Exact Crossing Minimization in Storyline Drawings

Storyline drawings are a popular visualization of interactions of a set of characters over time, e.g., to show participants of scenes in a book or movie. Characters are represented as $x$-monotone curves that converge vertically for interactions and diverge otherwise. Combinatorially, the task of computing storyline drawings reduces to finding a sequence of permutations of the character curves for the different time points, with the primary objective being crossing minimization of the induced character trajectories. In this paper, we revisit exact integer linear programming (ILP) approaches for this NP-hard problem. By enriching previous formulations with additional problem-specific insights and new heuristics, we obtain exact solutions for an extended new benchmark set of larger and more complex instances than had been used before. Our experiments show that our enriched formulations lead to better performing algorithms when compared to state-of-the-art modelling techniques. In particular, our best algorithms are on average 2.6-3.2 times faster than the state-of-the-art and succeed in solving complex instances that could not be solved before within the given time limit. Further, we show in an ablation study that our enrichment components contribute considerably to the performance of the new ILP formulation.

cs.DS↗
arXiv2024

SAT Encoding of Partial Ordering Models for Graph Coloring Problems

In this paper, we suggest new SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal "distance" between the assigned colors, and the goal is to minimize the "largest" color used. For the widely studied GCP, we experimentally compare our new SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.

cs.AI↗
arXiv2023

A Higher-Order Temporal H-Index for Evolving Networks

The H-index of a node in a static network is the maximum value $h$ such that at least $h$ of its neighbors have a degree of at least $h$. Recently, a generalized version, the $n$-th order H-index, was introduced, allowing to relate degree centrality, H-index, and the $k$-core of a node. We extend the $n$-th order H-index to temporal networks and define corresponding temporal centrality measures and temporal core decompositions. Our $n$-th order temporal H-index respects the reachability in temporal networks leading to node rankings, which reflect the importance of nodes in spreading processes. We derive natural decompositions of temporal networks into subgraphs with strong temporal coherence. We analyze a recursive computation scheme and develop a highly scalable streaming algorithm. Our experimental evaluation demonstrates the efficiency of our algorithms and the conceptional validity of our approach. Specifically, we show that the $n$-th order temporal H-index is a strong heuristic for identifying super-spreaders in evolving social networks and detects temporally well-connected components.

cs.SI↗
arXiv2022

TGLib: An Open-Source Library for Temporal Graph Analysis

We initiate an open-source library for the efficient analysis of temporal graphs. We consider one of the standard models of dynamic networks in which each edge has a discrete timestamp and transition time. Recently there has been a massive interest in analyzing such temporal graphs. Common computational data mining and analysis tasks include the computation of temporal distances, centrality measures, and network statistics like topological overlap, burstiness, or temporal diameter. To fulfill the increasing demand for efficient and easy-to-use implementations of temporal graph algorithms, we introduce the open-source library TGLib, which integrates efficient data structures and algorithms for temporal graph analysis. TGLib is highly efficient and versatile, providing simple and convenient C++ and Python interfaces, targeting computer scientists, practitioners, students, and the (temporal) network research community.

cs.DS↗
arXiv2022

A Temporal Graphlet Kernel for Classifying Dissemination in Evolving Networks

We introduce the \emph{temporal graphlet kernel} for classifying dissemination processes in labeled temporal graphs. Such dissemination processes can be spreading (fake) news, infectious diseases, or computer viruses in dynamic networks. The networks are modeled as labeled temporal graphs, in which the edges exist at specific points in time, and node labels change over time. The classification problem asks to discriminate dissemination processes of different origins or parameters, e.g., infectious diseases with different infection probabilities. Our new kernel represents labeled temporal graphs in the feature space of temporal graphlets, i.e., small subgraphs distinguished by their structure, time-dependent node labels, and chronological order of edges. We introduce variants of our kernel based on classes of graphlets that are efficiently countable. For the case of temporal wedges, we propose a highly efficient approximative kernel with low error in expectation. We show that our kernels are faster to compute and provide better accuracy than state-of-the-art methods.

cs.SI↗
arXiv2022

Strengthened Partial-Ordering Based ILP Models for the Vertex Coloring Problem

The vertex coloring problem asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. For this NP-hard problem, a variety of integer linear programming (ILP) models have been suggested. Among them, the assignment based and the partial-ordering based ILP models are attractive due to their simplicity and easy extendability. Moreover, on sparse graphs, both models turned out to be among the strongest exact approaches for solving the vertex coloring problem. In this work, we suggest additional strengthening constraints for the partial-ordering based ILP model, and show that they lead to improved lower bounds of the corresponding LP relaxation. Our computational experiments confirm that the new constraints are also leading to practical improvements. In particular, we are able to find the optimal solution of a previously open instance from the DIMACS benchmark set for vertex coloring problems

cs.DM↗
arXiv2022

Minimum-Error Triangulations for Sea Surface Reconstruction

We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al.~(Computers \& Geosciences, 157:104920, 2021) who have suggested to learn a triangulation $D$ of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by $D$ to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay ($k$-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in $\mathbb{R}^2$ with elevations: a set $\mathcal{S}$ that is to be triangulated, and a set $\mathcal{R}$ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in $\mathcal{R}$ to the triangulated surface that is obtained by interpolating elevations of $\mathcal{S}$ linearly in each triangle. Our goal is to find the triangulation of $\mathcal{S}$ that has minimum error with respect to $\mathcal{R}$. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless $P=NP$. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to $k$-OD triangulations for $k\le 7$. In particular, instances for which the number of connected components of the so-called $k$-OD fixed-edge graph is small can be solved within few seconds.

cs.CG↗
arXiv2022

Temporal Walk Centrality: Ranking Nodes in Evolving Networks

We propose the Temporal Walk Centrality, which quantifies the importance of a node by measuring its ability to obtain and distribute information in a temporal network. In contrast to the widely-used betweenness centrality, we assume that information does not necessarily spread on shortest paths but on temporal random walks that satisfy the time constraints of the network. We show that temporal walk centrality can identify nodes playing central roles in dissemination processes that might not be detected by related betweenness concepts and other common static and temporal centrality measures. We propose exact and approximation algorithms with different running times depending on the properties of the temporal network and parameters of our new centrality measure. A technical contribution is a general approach to lift existing algebraic methods for counting walks in static networks to temporal networks. Our experiments on real-world temporal networks show the efficiency and accuracy of our algorithms. Finally, we demonstrate that the rankings by temporal walk centrality often differ significantly from those of other state-of-the-art temporal centralities.

cs.SI↗
arXiv2021

An Index for Single Source All Destinations Distance Queries in Temporal Graphs

Temporal closeness is a generalization of the classical closeness centrality measure for analyzing evolving networks. The temporal closeness of a vertex $v$ is defined as the sum of the reciprocals of the temporal distances to the other vertices. Ranking all vertices of a network according to the temporal closeness is computationally expensive as it leads to a single-source-all-destination (SSAD) temporal distance query starting from each vertex of the graph. To reduce the running time of temporal closeness computations, we introduce an index to speed up SSAD temporal distance queries called Substream index. We show that deciding if a Substream index of a given size exists is NP-complete and provide an efficient greedy approximation. Moreover, we improve the running time of the approximation using min-hashing and parallelization. Our evaluation with real-world temporal networks shows a running time improvement of up to one order of magnitude compared to the state-of-the-art temporal closeness ranking algorithms.

cs.DB↗