We present our approach for the CG:SHOP 2026 challenge. In this international challenge, the goal was to find a median triangulation for a set of triangulations in the parallel flip reconfiguration graph of all triangulations of an underlying point set. Our simulated-annealing-based approach makes use of two ingredients: a heuristic edge selection for approximating the parallel flip distance of two given triangulations, and a heuristic procedure to generate good initial triangulations.

This repository contains benchmark instances for (s,t)-max-flow/min-cut, which were submitted to the 13th DIMACS Implementation Challenge. The instances are derived from the bicriteria polygon aggregation problem, which is studied in the following publications: Bicriteria Shapes: Hierarchical Grouping and Aggregation of Polygons with an Efficient Graph-Cut Approach. Peter Rottmann, Anne Driemel, Herman Haverkort, Heiko Röglin, Jan-Henrik Haunert. In: ACM Transactions on Spatial Algorithms and Systems, vol. 11(1), ACM, pages 3:1--3:23, 2025. A Simpler Approach for Monotone Parametric Minimum Cut: Finding the Breakpoints in Order. Arne Beines, Michael Kaibel, Philip Mayer, Petra Mutzel, Jonas Sauer. In: Proceedings of the 27th Workshop on Algorithm Engineering and Experiments (ALENEX'25), SIAM, pages 29--41, 2025. Bicriteria Polygon Aggregation with Arbitrary Shapes. Lotte Blank, David Eppstein, Jan-Henrik Haunert, Herman Haverkort, Benedikt Kolbe, Philip Mayer, Petra Mutzel, Alexander Naumann, Jonas Sauer. To appear in: Proceedings of the 34th Annual European Symposium on Algorithms (ESA'26), Leibniz International Proceedings in Informatics, 2026. Background These instances are derived from a real-world application: polygon aggregation for map simplification. Given is a set $P$ of building footprints, represented as 2D polygons. The objective is to find a set $S$ of interior-disjoint representative regions, such that each polygon in $P$ is fully contained in a region of $S$. We are interested in a solution that minimizes the objective function $g_\alpha(S) = A(S) + \alpha \cdot P(S)$, where $A(S)$ and $P(S)$ are the total area and perimeter of the regions in $S$, respectively. The parameter $\alpha$ controls the trade-off between faithfulness to the input (represented by the area) and shape simplicity (represented by the perimeter). In a cartographic application, it can be thought of as the "zoom factor" -- the further we zoom out, the simpler we want the shapes to become. We distinguish between two variants of the problem, both for a fixed choice of $\alpha$: Subdivision-based: A subdivision $D$ of the plane (e.g., a constrained Delaunay triangulation) is supplied in advance, such that each polygon in $P$ appears as a cell of $D$. The solution must be constructed by selecting cells from $D$ to add to $P$. The problem is solved via a transformation to (s,t)-min-cut on an augmented geometric dual of $D$ (see any of the papers listed above for details). Unrestricted: No restrictions are made regarding the shape of $S$ -- the only conditions are that $P$ must be covered and that the objective function $g_\alpha(S)$ is minimized. Blank et al. show that in this variant, the polygons are connected by circular arcs of radius $\alpha$ that fulfill several other conditions. The arcs are chosen from a set of $O(n^2)$ candidates. The problem can then be solved optimally via a transformation to the subdivision-based case, where the subdivision $D$ is created by superimposing all $O(n^2)$ arcs. This yields a solution in polynomial time, although the subdivision is much more complex than the constrained Delaunay triangulation. The resulting instances have some similarities with grid-based computer vision max-flow instances, which are built using a similar geometric-dual construction: They are sparse and have short (s,t)-paths. However, unlike typical vision instances, they do not have a regular structure because they are derived from human settlement areas. Consequently, although all nodes (except for s and t) have low degrees, the degrees are not entirely uniform. For the unrestricted variant, the graph is highly detailed because it is derived from a geometric intersection process between many circular arcs. As a side note, a parametric version of the problem, where $\alpha$ is not fixed, has also been studied. Here, the objective is to find an optimal solution for every possible value of $\alpha$. Beines et al. show that, with an equivalent reformulation of the objective function $g_\alpha$, this is a monotone parametric min-cut problem. The instances in this dataset are not parametric, but parametric instances can be found here. Contents The dataset is split into two groups, depending on how the subdivision was built: triangulations: Using a constrained Delaunay triangulation, as proposed by Rottmann et al. The instances are cities of varying sizes (Bonn, Cologne, Berlin, Miami) and the entire German state of Saarland. For each instance, there are five copies, with the different $\alpha$ values 100, 500, 1000, 5000, and 25000. Note that the graph structure is the same for all copies; only the weights are different. arcs: Using the geometric intersection of the candidate arcs, as proposed by Blank et al. for the unrestricted variant. The instances represent the towns of Ahrem, Edendorf, Friesheim and Gerolstein in the German state of North Rhine-Westphalia. For each instance, there are four copies, with the different $\alpha$ values 100, 500, 1000, 5000. Note that for this variant, the graph size increases dramatically with $\alpha$. The arc capacities represent a weighted tradeoff between area and perimeter, measured in square decimeters (dm^2) and rounded to the nearest integer. The $\alpha$ parameter is also measured in decimeters. In the arcs instances, this corresponds to the radii of the circular arcs from which the subdivision is formed, e.g., $\alpha=5000$ represents arcs with radii of 500m. Data Sources Ahrem, Friesheim, Edendorf, Gerolstein, Bonn, Cologne: OpenStreetMap data from Geofabrik Saarland: OpenStreetMap data from Geofabrik Berlin, Miami: GHS-OBAT project Format The files follow the format from the first DIMACS implementation challenge. This is a text format in which each line is prefixed with a character that specifies the type of line. Lines starting with c are comments and should be ignored. The first non-comment line is the problem line: p max NODES ARCS Here, max is the problem type (max-flow/min-cut), NODES is the number of nodes in the network, and ARCS is the number of directed arcs. This is followed by two node descriptor lines:n IDT tn IDS s Here, IDT is the id of the sink node and IDS is the id of the source node. Note that in this format, node ids start at 1. Finally, there is an arc descriptor line for every directed arc in the network: a SRC DST C This specifies an arc from node SRC to DST with capacity C. Capacities with values of int32_max or more should be interpreted as infinite. Note that the format does not require that a reverse arc exists for every directed arc. If reverse arcs are required by your algorithm, you must ensure that missing arcs are added with capacity 0. Credits and Contact This dataset was created by two research groups at the University of Bonn: the geoinformation group headed by Prof. Dr. Jan-Henrik Haunert and the Computational Analytics group headed by Prof. Dr. Petra Mutzel. It is released under the MIT license. When using it, please cite the publications listed above. If you want to report problems or give feedback on the dataset, please contact Jonas Sauer ([email protected]).

Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.

In the (continuous) Euclidean k -center problem, given n points in ℝ d and an integer k , the goal is to find k center points in ℝ d that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. Parameterized by k : Assuming the Exponential Time Hypothesis (ETH), we show that there is no f ( k ) n o ( k 1−1/ d ) -time algorithm for the Euclidean k -center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any (1+ε)-approximation algorithm running in time ( k /ε) o ( k 1−1/ d ) n O (1) , thereby establishing near-optimality of the corresponding approximation scheme by the same authors. Small k : Assuming the 3-SUM hypothesis, we prove that for any ε>0 there is no O ( n 2−ε )-time algorithm for the Euclidean 2-center problem in ℝ 3 . This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any ε > 0, the Euclidean 6-center problem in ℝ 2 also admits no O ( n 2−ε )-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.

The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of $\mathcal{O}(n^{8/3} log^3 n)$. To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of $\mathcal{O}(n^{8/3} \log^3 n)$ for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of $\tilde{\mathcal{O}}(n^4)$. Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.

Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including max-matching, max-spanning tree, as well as various measures for dataset diversity. For these problems, we show that the effect of dimension reduction is intimately tied to the *doubling dimension* $\lambda_X$ of the underlying dataset $X$---a quantity measuring intrinsic dimensionality of point sets. Specifically, the dimension required is $O(\lambda_X)$, which we also show is necessary for some of these problems. This is in contrast to classical dimension reduction results, whose dependence grow with the dataset size $|X|$. We also provide empirical results validating the quality of solutions found in the projected space, as well as speedups due to dimensionality reduction.

Abstract The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set $$\widetilde{P}$$ P ~ of lifted points in the hyperbolic plane. Since $$\widetilde{P}$$ P ~ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $$12g-6$$ 12 g - 6 with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter and the genus of the surface.