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Sparsifying Disk Intersection Graphs for Reliable Connectivity
The intersection graph induced by a set of n disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of vertices. In this work, we present a randomized sparsification algorithm that maintains connectivity between two regions in the computed graph, if the original graph remains "well-connected" even after removing an arbitrary `"attack'" set from both graphs. Thus, the new sparse graph has similar reliability to the original disk graph, and can withstand catastrophic failure of nodes while still providing a connectivity guarantee for the remaining graph. The new graph has near linear complexity, and can be constructed in near-linear time. The algorithm extends to any collection of shapes in the plane with near linear union complexity.
Copyright (c) 2024 Eliot Robson, Sariel Har-Peled https://creativecommons.org/licenses/by/4.0
Approximating the Directed Hausdorff Distance
The Hausdorff distance is a metric commonly used to compute the set similarity of geometric sets. For sets containing a total of n points, the exact distance can be computed naively in O(n2) time. In this paper, we show how to preprocess point sets individually so that the Hausdorff distance of any pair can then be approximated in linear time. We assume that the metric is doubling. The preprocessing time for each set is O(n log D) where D is the ratio of the largest to smallest pairwise distances of the input. In theory, this can be reduced to O(n log n) time using a much more complicated algorithm. We compute (1+eps)-approximate Hausdorff distance in (2+1/eps)O(d)n time in a metric space with doubling dimension d. The k-partial Hausdorff distance ignores k outliers to increase stability. Additionally, we give a linear-time algorithm to compute directed k$partial Hausdorff distance for all values of k at once with no change to the preprocessing.
Copyright (c) 2025 Oliver Chubet, Parth Parikh, Donald Sheehy, Siddharth Sheth https://creativecommons.org/licenses/by/4.0
Convex Covering Using Collections of Convex Polygons and Set Cover
In the convex covering problem, we are given a convex polygon with holes P and the goal is to cover P using a small number of convex polygons that lie inside P. In this paper, we solve the problem using the following strategy. We find a big collection of large (often maximal) convex polygons inside P and then solve several set cover problems to find a small subset of the collection that covers the whole polygon. The quality of our heuristics is confirmed by winning the second place in the CG:SHOP 2023 Challenge.
Copyright (c) 2026 Guilherme da Fonseca https://creativecommons.org/licenses/by/4.0
This is the guest editors' foreword for the special issue on selected papers from the 39th European Workshop on Computational Geometry, EuroCG'23.
Copyright (c) 2023 Clemens Huemer, Carlos Seara https://creativecommons.org/licenses/by/4.0
Arrangements of Pseudocircles: On Digons and Triangles
In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972). First, we discuss the maximum number of digons or touching points. Grünbaum conjectured that there are at most 2n - 2 digon cells or equivalently at most 2n - 2 touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of 2n - 2 touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Grünbaum conjectured that such arrangements have 2n - 4 triangles. Snoeyink and Hershberger (1991) established a lower bound of ⌈(4/3)n⌉. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only ⌈(16/11)n⌉ triangles. We provide a construction which shows that ⌈(4/3)n⌉ is the correct value.
Copyright (c) 2024 Stefan Felsner, Sandro Roch, Manfred Scheucher https://creativecommons.org/licenses/by/4.0
On the Crossing Number of Symmetric Configurations
For any finite set of points P in general position in the plane, we consider the drawing of the complete graph with vertex set P , whose edges are the straight-line segments joining pairs of points. The crossing number of P is the number of pairs of segments that intersect in their interior. The minimum crossing number over all such sets P of n points is known as the rectilinear or geometric crossing number of the complete graph Kn . The problem of determining this crossing number was posed by Erdős and Guy in the early 1970s and, although it has been extensively studied, it remains open to this date with exact values known only for n ≤ 27 and n = 30. In 2010, it was conjectured that whenever the number of points is a multiple of 3, there are crossing optimal point sets with 3-fold (triangular) rotational symmetry. Motivated by this conjecture, this paper provides lower and upper bounds for sym-crm (Kn ), the rectilinear crossing number of Kn when the minimum is restricted to m-fold symmetric configurations of points in the plane. Our bounds are tight for even symmetry.
Copyright (c) 2026 Bernardo M. Ábrego, Silvia Fernández-Merchant https://creativecommons.org/licenses/by/4.0
We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with convex drawings: outer $k$-planar graphs, where each edge is crossed by at most k other edges; and outer k-quasi-planar graphs, where no k edges can mutually cross. We show that the outer k-planar graphs are ⌊3.5 √k⌋-degenerate, and consequently that every √outer k-planar graph can be colored with ⌊3.5 √k⌋ + 1 colors. We further show that every outer k-planar graph has a balanced vertex separator of size at most 2k + 3. For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, e.g., this implies that none of these recognition problems is NP-hard unless the Exponential Time Hypothesis fails. We also show that the class of outer 3-quasi-planar graphs and the class of planar graphs are incomparable. Finally, we restrict outer k-planar and outer k-quasi-planar drawings to full drawings (where no crossing appears on the boundary of the outer face) and to closed drawings (where the vertex sequence on the boundary of the outer face is a Hamiltonian cycle in the graph). For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Since every outer k-planar graph has treewidth O(k), Courcelle’s theorem implies that closed outer k-planarity is linear-time testable. We leverage this result to further show that full outer k-planarity can also be tested in linear time.
Copyright (c) 2026 Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, Alexander Wolff https://creativecommons.org/licenses/by/4.0
Constructing Concise Convex Covers via Clique Covers
This work is the full version describing the winning implementation of the CG:SHOP 2023 Challenge. The topic of the Challenge was the convex cover problem: given a polygon P (with holes), find a minimum-cardinality set of convex polygons whose union equals P. We use a three-step approach: Create a suitable triangulation of P. Compute a visibility graph of the triangles. Solve a vertex clique cover problem on the visibility graph, from which we then derive the convex cover. This way we capture the geometric difficulty in the first step and the combinatorial difficulty in the third step.
Copyright (c) 2026 Mikkel Abrahamsen, William Bille Meyling, André Nusser https://creativecommons.org/licenses/by/4.0
Finding maximum matchings in RDV graphs efficiently
In this paper, we study the maximum matching problem in RDV graphs, i.e., vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a horizontal segment intersects one of a dynamically changing set of vertical segments, which in turn reduces to a range minimum query. Using a suitable data structure, we can therefore find a maximum matching in O(n log n) time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.
Copyright (c) 2026 Therese Biedl, Prashant Gokhale https://creativecommons.org/licenses/by/4.0
The Tripartite-Circle Crossing Number of Graphs With Two Small Partition Classes
A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of Ka,b,n, where a, b ≤ 2.
Copyright (c) 2024 Charles Camacho, Silvia Fernández-Merchant, Marija Jelić Milutinović, Rachel Kirsch, Linda Kleist, Elizabeth Bailey Matson, Jennifer White https://creativecommons.org/licenses/by/4.0
Outside-Obstacle Representations with All Vertices on the Outer Face
An obstacle representation of a graph G consists of a set of polygonal obstacles and a drawing of G as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.
Copyright (c) 2025 Oksana Firman, Philipp Kindermann, Jonathan Klawitter, Boris Klemz, Felix Klesen, Alexander Wolff https://creativecommons.org/licenses/by/4.0
Approximation Algorithms for Anchored Multiwatchman Routes
We study some variants of the k-Watchman Routes problem, the cooperative version of the classic Watchman Route problem in a simple polygon. The k watchmen may be required to see the whole polygon, or some pre-determined quota of area within the polygon, and we want to minimize the maximum length traveled by any watchman. While the single-watchman (k = 1$ version of the problem has received considerable attention and is relatively well understood, much less is known about the multiple watchmen (k >1) variant(s). We provide the first tight approximability results for the anchored k-Watchman Routes problem in a simple polygon, assuming k is fixed, by a fully-polynomial time approximation scheme. The basis for the FPTAS is provided by an exact dynamic programming algorithm. If k varies (i.e., it is part of the input), we give constant-factor approximations.
Copyright (c) 2025 Joseph S.B. Mitchell, Linh Nguyen https://creativecommons.org/licenses/by/4.0
Capturing the Shape of a Point Set With a Line Segment
Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line segment. Specifically, given a radius r, we say a line segment is representative of a point set P of n points if it is within distance r of each point p in P. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius r using the shortest line segment. We describe an algorithm to find the shortest representative segment in O(n log h + h log3 h) time, where h is the size of the convex hull of P. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in P move.
Copyright (c) 2025 Nathan van Beusekom, Marc van Kreveld, Max van Mulken, Marcel Roeloffzen, Bettina Speckmann, Jules Wulms https://creativecommons.org/licenses/by/4.0
The Erdős-Szekeres Maker-Breaker game is a two-player competitive game where both players alternately place points anywhere in the two dimensional Euclidean plane without overlapping, such that no three points are collinear. The first player (Maker) starts the game by placing her point and wants to obtain an empty convex polygon of a given size k such that the vertices of the polygon are chosen from these points and the second player (Breaker) wants to prevent it. We show that Maker wins the game for k <= 8. We also present a winning strategy for Maker for any k in general when Maker is allowed to place (1 + eps) times more points (each round on average) in comparison to Breaker, for any eps > 0. Further, we address the models of the game for equilateral empty convex polygons in the plane and empty convex polygons in square grids.
Copyright (c) 2026 Arun Kumar Das, Tomáš Valla https://creativecommons.org/licenses/by/4.0
This paper offers insights into the development and hosting of the Computational Geometry: Solving Hard Optimization Problems (CG:SHOP) Challenges. We examine the architecture of the supporting infrastructure, detail the tasks and workloads involved, discuss methods for selecting an effective set of instances, and share the lessons learned from the first seven iterations of the competition.
Copyright (c) 2025 Dominik Krupke, Phillip Keldenich, Rouven Kniep https://creativecommons.org/licenses/by/4.0
The Mutual Visibility Problem for Fat Robots with Lights
Given a set of n≥1 unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between them. This problem was solved by Sharma et al. [ICDCN, 2018] in the luminous robots model, where each robot is equipped with an externally visible light that can assume colors from a fixed set of colors, using 9 colors and O(n) rounds. In this work, we present an algorithm that requires only 2 colors and O(n) rounds. The number of colors is optimal since at least two colors are required even for pointrobots [Di Luna et al., Information and Computation, 2017].
Copyright (c) 2026 Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen https://creativecommons.org/licenses/by/4.0
Quadrangulating a Polygon with a Fixed Spirality using Steiner Points
Let P be an n-sided polygon on the plane, where n >= 4. A quadrangulation of P is a geometric plane graph obtained from P by adding edges to the interior of P so that every finite face is quadrilateral. It is easy to see that $P$ does not always admit a quadrangulation. So Ramaswami et al. introduced "Steiner points", which are auxiliary points added to P which helps quadrangulatability of P. Those points are said to be inner and outer if they are added to the interior and the exterior of P, respectively. They proved that every P with Steiner points added admits a quadrangulation, and estimated the number of those Steiner points by n, when all Steiner points are inner, and when all Steiner points are outer. On the other hand, Hidaka et al. used the notion of "spirality of P" which measures how far P is from being convex, and considered how a polygon P of spirality k admits a quadrangulation with inner Steiner points. In this paper, we consider the quadrangulatability of a polygon of P of spirality k when all Steiner points are outer, and when both inner and outer Steiner points are allowed.
Copyright (c) 2025 Fumiya Hidaka, Atsuhiro Nakamoto, Daiki Takahashi https://creativecommons.org/licenses/by/4.0
The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve p is c-packed if the length of the curve lying inside any ball is at most c times the radius of the ball, and its congestion is the minimum c for which pi is c-packed. This paper presents a randomized 42-approximation algorithm for computing the congestion of a curve (or any set of segments in the plane). It runs in O(n log2 n) time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm, which runs in (roughly) O(n4/3) time. We carefully combine new ideas with known techniques to obtain our new near-linear time algorithm.
Copyright (c) 2025 Sariel Har-Peled, Timothy Zhou https://creativecommons.org/licenses/by/4.0
An Improved Algorithm for Shortest Paths in Weighted Unit-Disk Graphs
Let V be a set of n points in the plane. The unit-disk graph G = (V, E) has vertex set V and an edge e_uv in E between vertices u, v in V if the Euclidean distance between u and v is at most 1. The weight of each edge e_uv is the Euclidean distance between u and v. Given V and a source point s in V we consider the problem of computing shortest paths in G from s to all other vertices. The previously best algorithm for this problem runs in O(n log2 n) time [Wang and Xue, SoCG'19]. The problem has an Omega(n log n) lower bound under the algebraic decision tree model. In this paper, we present an improved algorithm of O(n log2 n / log log n) time (under the standard real-RAM model). Furthermore, we show that the problem can be solved using O(n log n) comparisons under the algebraic decision tree model, matching the Omega(n log n) lower bound.
Copyright (c) 2026 Bruce W. Brewer, Haitao Wang https://creativecommons.org/licenses/by/4.0
This is the guest editors' foreword for the special issue on selected papers from the 34th Canadian Conference on Computational Geometry (CCCG 2022).
Copyright (c) 2023 Yeganeh Bahoo, Konstantinos Georgiou https://creativecommons.org/licenses/by/4.0
Minimum Coverage by Convex Polygons: The CG:SHOP Challenge 2023
We give an overview of the 2023 Computational Geometry Challenge targeting the problem Minimum Coverage by Convex Polygons, which consists of covering a given polygonal region (possibly with holes) by a minimum number of convex subsets, a problem with a long-standing tradition in Computational Geometry.
Copyright (c) 2026 Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Stefan Schirra https://creativecommons.org/licenses/by/4.0
A bipartite graph G = (X ∪ Y, E) is a 2-layer k-planar graph if it admits a drawing on the plane such that the vertices in X and Y are placed on two parallel lines respectively, edges are drawn as straight-line segments, and every edge involves at most k crossings. Angelini, Da Lozzo, Förster, and Schneck [GD 2020; Comput. J., 2024] showed that every 2-layer k-planar graph has pathwidth at most k + 1. In this paper, we show that this bound is sharp by giving a 2-layer k-planar graph with pathwidth k + 1 for every k ≥ 0. This improves their lower bound of (k + 3) / 2.
Copyright (c) 2026 Yuto Okada https://creativecommons.org/licenses/by/4.0
This is the guest editors' foreword for the special issue on the Fifth CG Challenge, CG:SHOP 2023.
Copyright (c) 2025 Sandor Fekete, Stefan Schirra https://creativecommons.org/licenses/by/4.0
This is the guest editors' foreword for the special issue on selected papers from the 35th Canadian Conference on Computational Geometry, CCCG 2023.
Copyright (c) 2025 Denis Pankratov, Jeff M. Phillips https://creativecommons.org/licenses/by/4.0
A Linear Time Algorithm for the Flip Distance between Plane Spanning Paths in Convex Point Sets
We provide a linear time algorithm to determine the flip distance between two plane spanning paths on a point set in convex position. At the same time, we show that the happy edge property does not hold in this setting. This has to be seen in contrast to several results for reconfiguration problems where the absence of the happy edge property implies algorithmic hardness of the flip distance problem. Further, we show that our algorithm can be adapted for (1) compatible flips (2) local flips and (3) flips for plane spanning paths in simple polygons.
Copyright (c) 2026 Oswin Aichholzer, Joseph Dorfer https://creativecommons.org/licenses/by/4.0
Dynamic Convex Hulls under Window-Sliding Updates
We consider the problem of dynamically maintaining the convex hull of a set S of points in the plane under the following special sequence of insertions and deletions (called window-sliding updates): insert a point to the right of all points of S and delete the leftmost point of S. We propose an O(|S|)-space data structure that can handle each update in O(1) amortized time, such that standard binary-search-based queries on the convex hull of S can be answered in O(log h) time, where h is the number of vertices of the convex hull of S, and the convex hull itself can be output in O(h) time.
Copyright (c) 2025 Haitao Wang https://creativecommons.org/licenses/by/4.0
On the Budgeted Hausdorff Distance Problem
Given a set P of n points in the plane, and a parameter k, we present an algorithm, whose running time is O(n3/2 √k log3/2n + kn log2n), with high probability, that computes a subset Q* of P of k points, that minimizes the Hausdorff distance between the convex-hulls of Q* and P . This is the first subquadratic algorithm for this problem if k is small.
Copyright (c) 2025 Sariel Har-Peled, Benjamin Raichel https://creativecommons.org/licenses/by/4.0
Every Combinatorial Polyhedron Can Unfold with Overlap
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net, a non-overlapping planar polygon. One can view his result as establishing that every combinatorial polyhedron P has a metric realization P that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of “reverse”): Is there a combinatorial polyhedron P such that, for every metric realization P in ℝ3, and for every spanning cut-tree T of the 1-skeleton, P cut by T unfolds to a net? In this paper we prove the answer is NO: Every combinatorial polyhedron has a realization and a cut-tree that edge-unfolds the polyhedron with overlap.
Copyright (c) 2025 Joseph O'Rourke https://creativecommons.org/licenses/by/4.0
We propose a label poisoning attack on geometric data sets against k-nearest neighbor classification. We provide an algorithm that can compute an εn-additive approximation of the optimal poisoning in n 22^{O(d+k/\ε)} time for a given data set X in R⁠d, where |X| = n. Our algorithm achieves its objectives through the application of multi-scale random partitions.
Thiruvenkadam Sivaprakasam Radhakrishnan
Copyright (c) 2025 Diego Ihara Centurion, Karine Chubarian, Bohan Fan, Francesco Sgherzi, Thiruvenkadam Sivaprakasam Radhakrishnan, Anastasios Sidiropoulos, Angelo Straight https://creativecommons.org/licenses/by/4.0
We introduce the notion of an eps-cover for a kernel range space. A kernel range space concerns a set of points X in Rd and the space of all queries by a fixed kernel (e.g., a Gaussian kernel K(p,.) = exp(-|p-.|2), where p is in \Rd). For a point set X of size n, a query returns a vector of values Rp in Rn, where the i-th coordinate (Rp)i = K(p, xi)$ for xi in X$ An eps-cover is a subset of points Q in Rd so for any p in Rd$ that (1/n) |Rp - Rq|1 <= eps for some q in Q. This is a smooth analog of Haussler's notion of eps-covers for combinatorial range spaces (e.g., defined by subsets of points within a ball query) where the resulting vectors Rp are in {0, 1}n instead of [0, 1]n. The kernel versions of these range spaces show up in data analysis tasks where the coordinates may be uncertain or imprecise, and hence one wishes to add some flexibility in the notion of inside and outside of a query range. Our main result is that, unlike combinatorial range spaces, the size of kernel eps-covers is independent of the input size n and dimension d. We obtain a bound of 2O~(1/\eps^2), where O~(f(1 / eps)) hides log factors in (1 / eps) that can depend on the kernel. This implies that by relaxing the notion of boundaries in range queries, eventually the curse of dimensionality disappears, and may help explain the success of machine learning in very high-dimensions.We also complement this result with a lower bound of almost (1/\eps)Omega(1/\eps), showing the exponential dependence on 1/eps is necessary.
Copyright (c) 2025 Jeff M. Phillips, Hasan Pourmahmoodaghababa https://creativecommons.org/licenses/by/4.0