Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Joseph S. B. Mitchell
We give an overview of the 2020 Computational Geometry Challenge, which targeted the problem of partitioning the convex hull of a given planar point set P into the smallest number of convex faces, such that no point of P is contained in the interior of a face.
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We give an overview of the 2022 Computational Geometry Challenge targeting the problem Minimum Partition into Plane Subsets, which consists of partitioning a given set of line segments into a minimum number of non-crossing subsets.
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.
We give an overview of the 2024 Computational Geometry Challenge targeting the problem \textsc{Maximum Polygon Packing}: Given a convex region $P$ in the plane, and a collection of simple polygons $Q_1, \ldots, Q_n$, each $Q_i$ with a respective value $c_i$, find a subset $S \subseteq \{1, \ldots,n\}$ and a feasible packing within $P$ of the polygons $Q_i$ (without rotation) for $i \in S$, maximizing $\sum_{i \in S} c_i$. Geometric packing problems, such as this, present significant computational challenges and are of substantial practical importance.
A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition based on the minimum number of internal faces. The problem has been shown to be NP-Hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear …
Central Triangulation under Parallel Flip Operations: The CG:SHOP Challenge 2026
研究在点集三角剖分中寻找基于并行翻转操作的中心三角剖分问题，提出通过并行翻转距离求解最优三角剖分的方法。
Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bo…
We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an O(log OPT)-approximation running in O(n^8)-time, where OPT denotes the minimum number of convex faces needed, to an O(sqrt(n) log n)-approximation algorithm running in O(n^2)-time. We say that a point set is k-directed if the (straight) lines containing at least thr…
The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map -- which represents, say, a geographical area -- into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (MCPP). A convex partition of a point set $P$ in the plane is a sub…
Partitioning Regular Polygons into Circular Pieces I: Convex Partitions
本文研究了正多边形的最优凸划分问题，证明等边三角形的最佳划分有无限多个部分，而正方形的最优长宽比差距缩小至0.01082。
Greedy Monochromatic Island Partitions
本文研究平面上彩色点集的最小基数单色岛划分问题，提出三种贪心构造算法并分析其近似比。
A note on non-crossing path partitions in the plane
本文通过计数凸位置点集的不相交路径划分，证明了双链点集的多边形化数量上界为 O(5.61^n)，并给出了若干新的上下界结果。
CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments. Surprisingly, the top three teams used the same technique, called conflict optimization. This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem. In this paper, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP). Then, the top three teams describe their different implementations of the same underlying strategy. We …
Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction of Tverberg's theorem from the special case k=2. We dash the hopes of a purely combinatorial deduction, but show that the case k=2 does imply that every set of O(k^2 log^2 k) points admits a Tverberg partition into k parts.
We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.
Let $P$ be a $d$-dimensional $n$-point set. A partition $T$ of $P$ is called a Tverberg partition if the convex hulls of all sets in $T$ intersect in at least one point. We say $T$ is $t$-tolerant if it remains a Tverberg partition after deleting any $t$ points from $P$. Soberón and Strausz proved that there is always a $t$-tolerant Tverberg partition with $\lceil n / (d+1)(t+1) \rceil$ sets. However, so far no nontrivial algorithms for computing or approximating such partitions have been presented. For $d \leq 2$, we show that the Soberón-Strausz bound can be improved, and we show how the cor…
Reconstructing a composition (union) of convex polytopes that perfectly fits the corresponding input point-cloud is a hard optimization problem with interesting applications in reverse engineering and rigid body dynamics simulations. We propose a pipeline that first extracts a set of planes, then partitions the input point-cloud into weakly convex clusters and finally generates a set of convex polytopes as the intersection of fitted planes for each partition. Finding the best-fitting convex polytopes is formulated as a combinatorial optimization problem over the set of fitted planes and is sol…
We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex par…
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