Discrete & Computational Geometry最新文献

The Parameterized Complexity of Coordinated Motion Planning. 协调运动规划的参数化复杂度。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-06-15 DOI: 10.1007/s00454-026-00849-3

Eduard Eiben, Robert Ganian, Iyad Kanj

{"title":"The Parameterized Complexity of Coordinated Motion Planning.","authors":"Eduard Eiben, Robert Ganian, Iyad Kanj","doi":"10.1007/s00454-026-00849-3","DOIUrl":"https://doi.org/10.1007/s00454-026-00849-3","url":null,"abstract":"In Coordinated Motion Planning (CMP), we are given a rectangular-grid on which k robots occupy k distinct starting gridpoints and need to reach k distinct destination gridpoints. In each time step, any robot may move to a neighboring gridpoint or stay in its current gridpoint, provided that it does not collide with other robots. The goal is to compute a schedule for moving the k robots to their destinations which minimizes a certain objective target-prominently the number of time steps in the schedule, i.e., the makespan, or the total length traveled by the robots. We refer to the problem arising from minimizing the former as CMP-M and the latter as CMP-L. Both CMP-M and CMP-L are fundamental problems that were posed as the computational geometry challenge of SoCG 2021, and CMP also embodies the famous <math><mrow><mo>(</mo> <msup><mi>n</mi> <mn>2</mn></msup> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </math> -puzzle as a special case. In this paper, we settle the parameterized complexity of CMP-M and CMP-L with respect to their two most fundamental parameters: the number of robots, and the objective target. We develop a new approach to establish the fixed-parameter tractability of both problems under the former parameterization that relies on novel structural insights into optimal solutions to the problem. When parameterized by the objective target, we show that CMP-L remains fixed-parameter tractable while CMP-M becomes para-NP-hard. The latter result is noteworthy, not only because it improves the previously-known boundaries of intractability for the problem, but also because the underlying reduction allows us to establish-as a simpler case-the NP-hardness of the classical Vertex Disjoint and Edge Disjoint Paths problems with constant path-lengths on grids.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"76 3","pages":"1786-1840"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13619704/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149082662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Minimal Dispersion on the Sphere. 球面上的最小色散。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-08-20 DOI: 10.1007/s00454-025-00812-8

Alexander E Litvak, Mathias Sonnleitner, Tomasz Szczepanski

{"title":"Minimal Dispersion on the Sphere.","authors":"Alexander E Litvak, Mathias Sonnleitner, Tomasz Szczepanski","doi":"10.1007/s00454-025-00812-8","DOIUrl":"10.1007/s00454-025-00812-8","url":null,"abstract":"The minimal spherical cap dispersion <math> <mrow> <msub><mrow><mspace></mspace> <mtext>disp</mtext> <mspace></mspace></mrow> <mi>C</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>)</mo></mrow> </mrow> </math> is the largest number <math><mrow><mi>ε</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo></mrow> </math> such that, for every n points on the d-dimensional Euclidean unit sphere <math> <msup><mrow><mi>S</mi></mrow> <mi>d</mi></msup> </math> , there exists a spherical cap with normalized area <math><mi>ε</mi></math> not containing any of these points. We study the behavior of <math> <mrow> <msub><mrow><mspace></mspace> <mtext>disp</mtext> <mspace></mspace></mrow> <mi>C</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>)</mo></mrow> </mrow> </math> as n and d grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on <math> <mrow> <msub><mrow><mspace></mspace> <mtext>disp</mtext> <mspace></mspace></mrow> <mi>C</mi></msub> <mrow><mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>)</mo></mrow> </mrow> </math> result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"76 3","pages":"1293-1321"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13619639/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149082686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Faithful Discretization of Verbose Directional Transforms. 详细方向变换的忠实离散化。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-25 DOI: 10.1007/s00454-025-00791-w

Brittany Terese Fasy, Samuel Micka, David L Millman, Anna Schenfisch, Lucia Williams

引用次数: 0

Some Fast Algorithms for Curves in Surfaces. 曲面中曲线的一些快速算法。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-04-21 DOI: 10.1007/s00454-026-00845-7

Marc Lackenby

{"title":"Some Fast Algorithms for Curves in Surfaces.","authors":"Marc Lackenby","doi":"10.1007/s00454-026-00845-7","DOIUrl":"https://doi.org/10.1007/s00454-026-00845-7","url":null,"abstract":"We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds <math><msub><mi>C</mi> <mn>1</mn></msub> </math> and <math><msub><mi>C</mi> <mn>2</mn></msub> </math> in a compact orientable surface S. The surface S is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of <math><msub><mi>C</mi> <mn>1</mn></msub> </math> and <math><msub><mi>C</mi> <mn>2</mn></msub> </math> . This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of S and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether <math><msub><mi>C</mi> <mn>1</mn></msub> </math> and <math><msub><mi>C</mi> <mn>2</mn></msub> </math> are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"76 1","pages":"539-588"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13291048/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148334139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Morse Theory for the k-NN Distance Function. k-NN距离函数的Morse理论。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-19 DOI: 10.1007/s00454-025-00795-6

Yohai Reani, Omer Bobrowski

引用次数: 0

A Full Halin Grid Theorem. 一个完整的Halin网格定理。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2024-12-09 DOI: 10.1007/s00454-024-00707-0

Agelos Georgakopoulos, Matthias Hamann

引用次数: 0

Maximum Betti Numbers of Čech Complexes. Čech配合物的最大贝蒂数。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-11-10 DOI: 10.1007/s00454-025-00796-5

Herbert Edelsbrunner, János Pach

{"title":"Maximum Betti Numbers of Čech Complexes.","authors":"Herbert Edelsbrunner, János Pach","doi":"10.1007/s00454-025-00796-5","DOIUrl":"https://doi.org/10.1007/s00454-025-00796-5","url":null,"abstract":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> and any radius satisfies <math> <mrow><msub><mi>β</mi> <mi>p</mi></msub> <mrow></mrow> <mo>=</mo> <mi>O</mi> <mrow><mo>(</mo> <msup><mi>N</mi> <mi>m</mi></msup> <mo>)</mo></mrow> </mrow> </math> , with <math><mrow><mi>m</mi> <mo>=</mo> <mo>min</mo> <mo>{</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mrow><mrow><mo>⌈</mo></mrow> <mi>d</mi> <mo>/</mo> <mn>2</mn> <mrow><mo>⌉</mo></mrow> </mrow> <mo>}</mo></mrow> </math> . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of <math><mrow><mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> </math> points in <math> <msup><mrow><mi>R</mi></mrow> <mn>3</mn></msup> </math> and two radii such that the first Betti number of the Čech complex at one radius is <math> <mrow> <msup><mrow><mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> <mo>-</mo> <mn>1</mn></mrow> </math> , and the second Betti number of the Čech complex at the other radius is <math><msup><mi>n</mi> <mn>2</mn></msup> </math> .","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 2","pages":"597-624"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12953301/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147357507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric Realizations of ν -associahedra via Brick Polyhedra. 用砖块多面体实现ν -共轭面体的几何形状。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-09-22 DOI: 10.1007/s00454-025-00766-x

Cesar Ceballos, Matthias Müller

引用次数: 0

Non-Homotopic Drawings of Multigraphs. 多重图的非同伦图。

IF 0.7 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2026-06-14 DOI: 10.1007/s00454-026-00851-9

António Girão, Freddie Illingworth, Alex Scott, David R Wood

{"title":"Non-Homotopic Drawings of Multigraphs.","authors":"António Girão, Freddie Illingworth, Alex Scott, David R Wood","doi":"10.1007/s00454-026-00851-9","DOIUrl":"https://doi.org/10.1007/s00454-026-00851-9","url":null,"abstract":"A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is k-crossing if every pair of edges (self-)intersects at most k times. We prove that the number of edges in an n-vertex non-homotopic k-crossing multigraph is at most <math><msup><mn>6</mn> <mrow><mn>13</mn> <mi>n</mi> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> </msup> </math> , which is a big improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, m, in such a drawing is at most <math><mrow><mn>2</mn> <mfenced> <mrow> <mtable> <mtr> <mtd><mrow><mn>2</mn> <mi>n</mi></mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn></mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math> and the number of crossings is <math><mrow><mi>Ω</mi> <mrow><mo>(</mo></mrow> <mfrac><msup><mi>m</mi> <mrow><mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi></mrow> </msup> <msup><mi>n</mi> <mrow><mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi></mrow> </msup> </mfrac> <mrow><mo>)</mo></mrow> </mrow> </math> . For fixed k these bounds are both best possible up to a constant multiplicative factor.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"76 3","pages":"1862-1885"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13619773/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149082730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the Size of Chromatic Delaunay Mosaics. 论彩色德劳内马赛克的尺寸。

IF 0.6 3区数学

Discrete & Computational Geometry Pub Date : 2026-01-01 Epub Date: 2025-09-30 DOI: 10.1007/s00454-025-00778-7

Ranita Biswas, Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, Morteza Saghafian

{"title":"On the Size of Chromatic Delaunay Mosaics.","authors":"Ranita Biswas, Sebastiano Cultrera di Montesano, Ondřej Draganov, Herbert Edelsbrunner, Morteza Saghafian","doi":"10.1007/s00454-025-00778-7","DOIUrl":"10.1007/s00454-025-00778-7","url":null,"abstract":"Given a locally finite set <math><mrow><mi>A</mi> <mo>⊆</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> and a coloring <math><mrow><mi>χ</mi> <mo>:</mo> <mi>A</mi> <mo>→</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>s</mi> <mo>}</mo></mrow> </math> , we introduce the chromatic Delaunay mosaic of <math><mi>χ</mi></math> , which is a Delaunay mosaic in <math> <msup><mrow><mi>R</mi></mrow> <mrow><mi>d</mi> <mo>+</mo> <mi>s</mi></mrow> </msup> </math> that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that d and s are constants. For example, if A is finite with <math><mrow><mi>n</mi> <mo>=</mo> <mrow><mo>#</mo> <mi>A</mi></mrow> </mrow> </math> , and the coloring is random, then the chromatic Delaunay mosaic has <math><mrow><mi>O</mi> <mo>(</mo> <msup><mi>n</mi> <mrow><mo>⌈</mo> <mi>d</mi> <mo>/</mo> <mn>2</mn> <mo>⌉</mo></mrow> </msup> <mo>)</mo></mrow> </math> cells in expectation. In contrast, for Delone sets and Poisson point processes in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> all colorings of a well spread set of n points have chromatic Delaunay mosaics of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 1","pages":"24-47"},"PeriodicalIF":0.6,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12748319/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145879476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0