computational geometry developers

Abstract: We give the first explicit constant for crossing families of linear size by showing that every set of $n$ points in general position in the plane contains a crossing family of size approximately $\frac{n}{79}$. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large different pattern, namely non-convex 4-clusters: We show that every point set contains either a large non-convex 4-cluster or a large crossing family. The second condition is the existence of many crossings: We give a bound for the size of a crossing family in terms of the number of crossings of the complete geometric graph drawn on the point set.

Abstract: Given a finite planar point set $X=\{x_1,\ldots,x_n\}$, let $a_1<a_2< \cdots < a_s$ denote the $s \leq {n \choose 2}$ different (Euclidean) distances between points in $X$. The aim is to retain as many of the points as possible such that certain imposed inter-pair distances are avoided. For example, we show that: (A) Given a set $X$ of $n$ points in the plane, an independent set of size $0.2293 n$ in the unit distance graph determined by $X$ can be found in $O(n^2)$ time. For any fixed $k \in \mathbf{N}$ there exists a constant $c_k>0$ such that for any set $X$ of $n$ points in the plane, and set $D$ of size $k$, there exists a subset $X' \subseteq X$ of size $|X'| \geq c_k |X|$ in which no pair of points is at some distance in $D$ (we say that $X'$ avoids distances in $D$). Such a set can be found efficiently. (B) (i) There is a subset of $X$ of size $n/2$ that avoids the maximum distance; this bound cannot be improved. (ii) There is a subset of $X$ of size $8n/47$ that avoids both the minimum and maximum distance; on the other hand, there exists $X$ such that at most $5n/16$ points can be retained with this property. (iii) There is a subset of $X$ of size $n/7$ whose diameter is at most half the original diameter; on the other hand, there exists $X$ such that at most $n/6$ points can be retained with this property. (C) (i) For any $\varepsilon>0$, a $(1-\varepsilon)$-approximation for \textsc{MIS} in the minimum distance graph of $n$ points in the plane can be found in $2^{O(1/\varepsilon)} n^{O(1)}$ time. (ii) Given a set $X$ of $n$ points in the plane, a maximum independent set in the graph of diameters can be computed in $O(n^2)$ time. The techniques used to handle the planar case can be extended to $\mathbf{R}^d$ for any fixed dimension $d$.

Abstract: The Preprocessed 3SUM problem asks to preprocess a set $A$ of $n$ numbers so that we can quickly decide whether a query subset $A' \subseteq A$ contains a solution $a+b=c$. The best known query bounds differ substantially between integer and real inputs. For integers, a series of advances with near-quadratic preprocessing reduced the query time from $\tilde O(n^{13/7})$ [Chan, Lewenstein; STOC '15] to $\tilde O(n^{11/6})$ [Chan, Vassilevska W., Xu; STOC '23], and finally to $\tilde O(n^{3/2})$ [Kasliwal, Polak, Sharma; SOSA '25]. For reals, only the first of these bounds has been achieved, by adapting the corresponding integer algorithm [Fischer; SODA '25]. Is this gap inherent to the more limited tools available for real inputs? We give a randomized real-to-integer reduction with $\tilde O(n^2)$ preprocessing overhead and no asymptotic increase in query time. This yields $\tilde O(n^2)$ preprocessing and $\tilde O(n^{3/2})$ query time for real inputs, closing the gap. We show how to extend our result to the setting where one set is unknown at preprocessing time. Our reduction is conceptually simple, and relies on an interesting technical toolkit around old and new tools from sparse linear algebra. More broadly, it suggests a route to real-to-integer reductions for other additive problems.

Abstract: We prove co-RE-completeness, and thus undecidability, of the following problem: given a single (connected) polycube, decide whether it tiles 3D Euclidean space by translations. We reduce from Wang tiling using the decorated two-prime Sudoku construction of Greenfeld and Tao and a cyclic encoding adapted from OpenAI's 3D aperiodic tile, and apply a reduction of Kim to make the prototile connected (via faces). Dimension three is optimal: translational monotiling is known to be decidable in $\mathbb{Z}^2$ and for a single (possibly disconnected) polyomino in $\mathbb{R}^2$.

Abstract: The computation of the vertices of a polyhedron described by a system of linear inequalities is a central problem in polyhedral computation. It is a fundamental step in the conversion between H-representations, by linear inequalities, and V-representations, by vertices and extreme rays. This operation plays an important role both in the study of polyhedra and their combinatorics in mathematics and in applications to software and system verification. We present a certificate-based approach for formally verifying the computation of the vertices of a polyhedron. Given an informally computed list of vertices, our method allows to certify in the proof assistant Rocq that the list is complete, or even exact. The cornerstone of the method is a new completeness criterion based on an abstract simplicial complex that generalizes a triangulation of the normal fan of the polyhedron. A significant advantage over previous approaches is that the usually expensive numerical computations are essentially reduced to membership tests to the polyhedron, while the other steps are cheap combinatorial tests. We implement the certification method and prove its correctness in the proof assistant Rocq. We experiment with it on a variety of polyhedra, including Birkhoff polytopes, cross-polytopes, cubes, permutahedra, hypersimplices, and high-dimensional polytopes involved in the disproof of the Hirsch conjecture. Our experiments show that certification with the Rocq-to-OCaml extracted checker is typically 1.5x to over 5x faster than vertex enumeration by the state-of-the-art informal C implementation lrslib of the reverse search method.

Abstract: A quadrilateral block decomposition of a planar domain is judged by whether it is complete, whether its elements are well shaped, and how many of its vertices are irregular. The last has a provable floor: the discrete Gauss-Bonnet identity enforces a lower bound on the total vertex irregularity of any all-quadrilateral mesh of a given domain purely based on its topology and corner angles. We train a reinforcement learning agent to build decompositions that reach this bound, which we call par. It acts directly on the mesh's half-edge data structure through local edits, with a policy network whose convolutions follow the mesh's own connectivity, so it applies unchanged to domains larger than any seen in training. The reward targets the floor directly, and it is sparse: random play reaches it on no domain with more than eight sides. We overcome this exploration barrier via behaviour cloning on optimal meshes that are trivial to construct, walked backward into demonstrations, before training it with PPO. On 96 held-out domains the agent produces an all-quadrilateral mesh on every one, a usable one on 95.7 on average, and a provably optimal one on 90; Gmsh's strongest configuration at the same element count completes 51, is usable on 38 and optimal on none, and even at three to fourteen times the elements never produces a more regular mesh. On 64 domains twice the training size the agent completes all, is usable on 62, and keeps a median excess over par below one against Gmsh's 39 at the same element count.

Abstract: In this paper, we provide a construction of an acyclic discrete vector field that is compatible with a total order associated with a lower star filtration on a simplicial complex, called the colex vector field. We show that the colex vector field induces a filtered acyclic vector field, whose resulting Morse complex computes the persistent homology of the lower star filtration on the underlying simplicial complex. We show that the colex vector field can be recomputed quickly when the vertex function that induces the lower star filtration is modified via order-adjacent vertex swaps. We provide a framework for storing and computing the number of paths between cells in the simplicial complex, as well as a method to update these values quickly when the colex vector field changes. Finally, we provide publicly available proof-of-concept code for the ideas shown. When applying it to the Persistent Homology Transform, we show that its runtime is comparable to a more standard matrix reduction approach.

Abstract: Packing and covering problems for geometric regions have been studied under many notions of complexity, including VC-dimension, union complexity, shallow-cell complexity, and fatness. Although these restrictions often yield constant-factor approximation algorithms, they do not by themselves generally lead to PTASs. A recurring feature of known hardness constructions is that one region may be \emph{pierced} by many others: a region $B$ pierces $A$ when $A\setminus B$ is disconnected. We study geometric instances through the \emph{piercing degree}. Since piercing is symmetric for Jordan regions, this is the maximum degree of the corresponding piercing graph. Our main result is that, for every fixed piercing degree, the standard local-search algorithms give PTASs for the unweighted \emph{Discrete Independent Set} and \emph{Set Cover} problems. The proof constructs a sublinear balanced separator for an appropriate locality graph and applies it adaptively throughout the recursive local-search analysis. This guarantee depends only on the piercing degree; in particular, it places no bound on the number of components created by an individual piercing pair. We also prove a polynomial shallow-trace bound depending only on the piercing degree. As consequences, for every fixed piercing degree, weighted Set Cover admits a deterministic $C_r$-approximation and weighted Discrete Independent Set admits a deterministic $O(r+1)$-approximation. These results extend the known guarantees for non-piercing families and apply, for example, to axis-parallel rectangles when every rectangle is pierced by only a bounded number of other rectangles.

Abstract: We study the problem of constructing metric spanners in general metric spaces in subquadratic time when given blackbox access to a fast algorithm for batch approximate nearest neighbor search. In particular, we show the following results for any metric space $\mathsf{M} = ([n], \mathsf{d})$ with aspect ratio $Δ$ admitting a $c$-approximate batch nearest neighbor search algorithm with runtime $τ_{\mathsf{M}}(n)$, (1) There exists an algorithm that, for any $k \in \mathbb{N}$, constructs an $O(c k)$-distortion spanner with $\tilde{O}(kn^{1+1/2k} \log Δ)$ edges and runs in time $\tilde{O}(τ_{\mathsf{M}} \cdot k n^{1/k} \log Δ)$. (2) Any algorithm that learns at most $o(n^{1+1/k}/k)$ pairwise distances by querying a distance oracle and a blackbox batch nearest neighbor search oracle necessarily incurs $Ω(c k)$ distortion. Our results entail that (truly) sub-quadratic time algorithms for spanner construction is equivalent to subquadratic time BANN (up to constant-factor losses). As a further application, we use our fast spanner constructions to obtain a fast algorithm for approximating the Wasserstein distance $\mathsf{W}_q$, for all $q > 1$, over any metric space admitting an efficient batch approximate nearest neighbor search algorithm. Together with recent new efficient algorithms for approximate nearest neighbor search in $\ell_p$ spaces, for $p > 2$, our results entail the first subquadratic time algorithms for spanner construction (with the stated size-distortion tradeoff) and $\mathsf{W}_q$ distance approximation over these metric spaces.

Abstract: Backman and Liu proved that every integral generalized permutohedron of type $A$, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type $B$: the delta-matroid simplex \[\operatorname*{conv}\{\mathbf{0},\ e_1+e_2,\ e_1+e_3,\ e_2+e_3\}\] has normalized volume $2$ and no lattice points other than its vertices, so it has no unimodular triangulation. We show moreover that, up to the natural symmetries of the $0/1$ cube and deletion of constant coordinates, it is the unique non-unimodular delta-matroid polytope that is a simplex. We prove instead that every delta-matroid polytope admits a regular dyadic triangulation, meaning a lattice triangulation whose maximal simplices have normalized volumes that are powers of two. More generally, every integral type $B$ generalized permutohedron admits such a triangulation. The main lattice-theoretic ingredient is that the type $B$ root configuration forms a totally dyadic system, a $2$-local analogue of total unimodularity. As a consequence, these polytopes satisfy a dyadic version of the integer decomposition property. In each dimension the corresponding exponent can be chosen uniformly, even though ordinary integer decomposition can fail for delta-matroid polytopes.

Abstract: In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).