Pub Date : 2024-12-16DOI: 10.1016/j.comgeo.2024.102159
Arthur Jansen, Bart Kuijpers
Space-time prisms provide a framework to model the uncertainty on the space-time points that a moving object may have visited between measured space-time locations, provided that a bound on the speed of the moving object is given. In this model, the alibi query asks whether two moving objects, given by their respective measured space-time locations and speed bound, may have met. An analytical solution to this problem was first given by Othman [15]. In this paper, we address the generalised alibi query that asks the same question for an arbitrary number of moving objects. We provide several solutions (mainly via the spatial and temporal projection) to this query with varying time complexities. These algorithmic solutions rely on techniques from convex and semi-algebraic geometry. We also address variants of the generalised alibi query where the question is asked for a given spatial location or a given moment in time.
{"title":"Geometric and algorithmic solutions to the generalised alibi query","authors":"Arthur Jansen, Bart Kuijpers","doi":"10.1016/j.comgeo.2024.102159","DOIUrl":"10.1016/j.comgeo.2024.102159","url":null,"abstract":"<div><div>Space-time prisms provide a framework to model the uncertainty on the space-time points that a moving object may have visited between measured space-time locations, provided that a bound on the speed of the moving object is given. In this model, the <em>alibi query</em> asks whether two moving objects, given by their respective measured space-time locations and speed bound, may have met. An analytical solution to this problem was first given by Othman <span><span>[15]</span></span>. In this paper, we address the <em>generalised alibi query</em> that asks the same question for an arbitrary number <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> of moving objects. We provide several solutions (mainly via the spatial and temporal projection) to this query with varying time complexities. These algorithmic solutions rely on techniques from convex and semi-algebraic geometry. We also address variants of the generalised alibi query where the question is asked for a given spatial location or a given moment in time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"127 ","pages":"Article 102159"},"PeriodicalIF":0.4,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.comgeo.2024.102151
Hugo A. Akitaya , Maike Buchin , Majid Mirzanezhad , Leonie Ryvkin , Carola Wenk
The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is “realizable”, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram?
Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in , showing -hardness. We use this to show that for curves in the realizability problem is -complete, both for continuous and discrete Fréchet distances. We prove that the continuous case in is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in we show that the problem becomes solvable in polynomial time.
{"title":"Realizability of free spaces of curves","authors":"Hugo A. Akitaya , Maike Buchin , Majid Mirzanezhad , Leonie Ryvkin , Carola Wenk","doi":"10.1016/j.comgeo.2024.102151","DOIUrl":"10.1016/j.comgeo.2024.102151","url":null,"abstract":"<div><div>The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is “<em>realizable</em>”, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram?</div><div>Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>2</mn></mrow></msup></math></span>, showing <span><math><mo>∃</mo><mi>R</mi></math></span>-hardness. We use this to show that for curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>≥</mo><mn>2</mn></mrow></msup></math></span> the realizability problem is <span><math><mo>∃</mo><mi>R</mi></math></span>-complete, both for continuous and discrete Fréchet distances. We prove that the continuous case in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> we show that the problem becomes solvable in polynomial time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"127 ","pages":"Article 102151"},"PeriodicalIF":0.4,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 10.1016/j.comgeo.2024.102150
Eunku Park, Antoine Vigneron
We give an embedding of the Poincaré halfspace into a discrete metric space based on a binary tiling of , with additive distortion . It yields the following results. We show that any subset P of n points in can be embedded into a graph-metric with vertices and edges, and with additive distortion . We also show how to construct, for any k, an -purely additive spanner of P with Steiner vertices and edges, where is the kth-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for P of size . It allows us to answer approximate near-neighbor queries in time, with additive error . These constructions can be done in time.
我们给出了一种基于二元平铺的离散度量空间 HD 的嵌入方法，其附加变形为 O(logD)。它产生了以下结果。我们证明，HD 中任何 n 个点的子集 P 都可以嵌入到一个具有 2O(D)n 个顶点和边的图度量空间中，其附加变形为 O(logD)。我们还展示了如何为任意 k 构建 P 的 O(klogD)-purely additive spanner，该 spanner 具有 2O(D)n 个 Steiner 顶点和 2O(D)n⋅λk(n) 条边，其中 λk(n) 是第 k 行逆阿克曼函数。最后，我们展示了如何为 P 构建大小为 2O(D)n 的近似 Voronoi 图。它允许我们在 2O(D)+O(Dlogn) 时间内回答近似近邻查询，加法误差为 O(logD)。这些构造可以在 2O(D)nlogn 时间内完成。
{"title":"Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces","authors":"Eunku Park, Antoine Vigneron","doi":"10.1016/j.comgeo.2024.102150","DOIUrl":"10.1016/j.comgeo.2024.102150","url":null,"abstract":"<div><div>We give an embedding of the Poincaré halfspace <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span> into a discrete metric space based on a binary tiling of <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span>, with additive distortion <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>D</mi><mo>)</mo></math></span>. It yields the following results. We show that any subset <em>P</em> of <em>n</em> points in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow></msup></math></span> can be embedded into a graph-metric with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span> vertices and edges, and with additive distortion <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>D</mi><mo>)</mo></math></span>. We also show how to construct, for any <em>k</em>, an <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mi>D</mi><mo>)</mo></math></span>-purely additive spanner of <em>P</em> with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span> Steiner vertices and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi><mo>⋅</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> edges, where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the <em>k</em>th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for <em>P</em> of size <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi></math></span>. It allows us to answer approximate near-neighbor queries in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mo>+</mo><mi>O</mi><mo>(</mo><mi>D</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time, with additive error <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mi>D</mi><mo>)</mo></math></span>. These constructions can be done in <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></msup><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi></math></span> time.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102150"},"PeriodicalIF":0.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 10.1016/j.comgeo.2024.102148
Ulrich Bauer , Abhishek Rathod
We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of . Our second result shows that Min-Morse Matching is -hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor ρ. The above hardness results are applicable to complexes of dimension ≥2.
On the positive side, we provide a factor approximation algorithm for Min-Morse Matching on 2-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa–Farber model of random complexes.
{"title":"Parameterized inapproximability of Morse matching","authors":"Ulrich Bauer , Abhishek Rathod","doi":"10.1016/j.comgeo.2024.102148","DOIUrl":"10.1016/j.comgeo.2024.102148","url":null,"abstract":"<div><div>We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of <span>Min-Morse Matching</span> within a factor of <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>log</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></mrow></msup><mo></mo><mi>n</mi></mrow></msup></math></span>. Our second result shows that <span>Min-Morse Matching</span> is <span><math><mi>W</mi><mo>[</mo><mi>P</mi><mo>]</mo></math></span>-hard with respect to the standard parameter. Next, we show that <span>Min-Morse Matching</span> with standard parameterization has no FPT approximation algorithm for <em>any</em> approximation factor <em>ρ</em>. The above hardness results are applicable to complexes of dimension ≥2.</div><div>On the positive side, we provide a factor <span><math><mi>O</mi><mo>(</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></mfrac><mo>)</mo></math></span> approximation algorithm for <span>Min-Morse Matching</span> on 2-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa–Farber model of random complexes.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102148"},"PeriodicalIF":0.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-29DOI: 10.1016/j.comgeo.2024.102149
Gerardo L. Maldonado, Edgardo Roldán-Pensado
Grünbaum's equipartition problem asked if for any measure μ on there are always d hyperplanes which divide into μ-equal parts. This problem is known to have a positive answer for and a negative one for . A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for and there is reason to expect it to have a negative answer for . In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of 8n in can be split evenly by 3 mutually orthogonal planes. To our surprise, it seems the probability that a random set of 8 points chosen uniformly and independently in the unit cube does not admit such a partition is less than 0.001.
{"title":"On the orthogonal Grünbaum partition problem in dimension three","authors":"Gerardo L. Maldonado, Edgardo Roldán-Pensado","doi":"10.1016/j.comgeo.2024.102149","DOIUrl":"10.1016/j.comgeo.2024.102149","url":null,"abstract":"<div><div>Grünbaum's equipartition problem asked if for any measure <em>μ</em> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> there are always <em>d</em> hyperplanes which divide <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> into <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span> <em>μ</em>-equal parts. This problem is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>3</mn></math></span> and a negative one for <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span>. A variant of this question is to require the hyperplanes to be mutually orthogonal. This variant is known to have a positive answer for <span><math><mi>d</mi><mo>≤</mo><mn>2</mn></math></span> and there is reason to expect it to have a negative answer for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>. In this note we exhibit measures that prove this. Additionally, we describe an algorithm that checks if a set of 8<em>n</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> can be split evenly by 3 mutually orthogonal planes. To our surprise, it seems the probability that a random set of 8 points chosen uniformly and independently in the unit cube does not admit such a partition is less than 0.001.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102149"},"PeriodicalIF":0.4,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.comgeo.2024.102147
Alexander Heaton , Matthias Himmelmann
Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of , and prove convergence results.
We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.
{"title":"Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization","authors":"Alexander Heaton , Matthias Himmelmann","doi":"10.1016/j.comgeo.2024.102147","DOIUrl":"10.1016/j.comgeo.2024.102147","url":null,"abstract":"<div><div>Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence <em>retraction maps</em> are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and prove convergence results.</div><div>We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"126 ","pages":"Article 102147"},"PeriodicalIF":0.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-13DOI: 10.1016/j.comgeo.2024.102135
Jaehoon Chung , Sang Won Bae , Chan-Su Shin , Sang Duk Yoon , Hee-Kap Ahn
We consider an optimization problem of inscribing a unit rectangle in a convex polygon. An axis-aligned unit rectangle is an axis-aligned rectangle whose horizontal sides are of length 1. A unit rectangle of orientation θ is a copy of an axis-aligned unit rectangle rotated by θ in counterclockwise direction. The goal is to find a largest unit rectangle inscribed in a convex polygon over all orientations in . This optimization problem belongs to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.
{"title":"Largest unit rectangles inscribed in a convex polygon","authors":"Jaehoon Chung , Sang Won Bae , Chan-Su Shin , Sang Duk Yoon , Hee-Kap Ahn","doi":"10.1016/j.comgeo.2024.102135","DOIUrl":"10.1016/j.comgeo.2024.102135","url":null,"abstract":"<div><p>We consider an optimization problem of inscribing a unit rectangle in a convex polygon. An axis-aligned unit rectangle is an axis-aligned rectangle whose horizontal sides are of length 1. A unit rectangle of orientation <em>θ</em> is a copy of an axis-aligned unit rectangle rotated by <em>θ</em> in counterclockwise direction. The goal is to find a largest unit rectangle inscribed in a convex polygon over all orientations in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></math></span>. This optimization problem belongs to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"124 ","pages":"Article 102135"},"PeriodicalIF":0.4,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000579/pdfft?md5=421deb79b0fe58ffb995ba93bffa3330&pid=1-s2.0-S0925772124000579-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-06DOI: 10.1016/j.comgeo.2024.102134
Thomas Fernique
A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.
{"title":"Packing unequal disks in the Euclidean plane","authors":"Thomas Fernique","doi":"10.1016/j.comgeo.2024.102134","DOIUrl":"10.1016/j.comgeo.2024.102134","url":null,"abstract":"<div><p>A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"124 ","pages":"Article 102134"},"PeriodicalIF":0.4,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000567/pdfft?md5=fb180e9154b1ec63995a4bc9108b1b08&pid=1-s2.0-S0925772124000567-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-22DOI: 10.1016/j.comgeo.2024.102124
Tomer Cohen, Ariel Kulik, Hadas Shachnai
We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.
Our main result is a -approximation algorithm for 2VMK, for every fixed , thus improving the best known ratio of which follows as a special case from a result of Fleischer et al. (2011) [6].
Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.
{"title":"Improved approximation for two-dimensional vector multiple knapsack","authors":"Tomer Cohen, Ariel Kulik, Hadas Shachnai","doi":"10.1016/j.comgeo.2024.102124","DOIUrl":"10.1016/j.comgeo.2024.102124","url":null,"abstract":"<div><p>We study the <span>uniform</span> 2<span>-dimensional vector multiple knapsack</span> (2VMK) problem, a natural variant of <span>multiple knapsack</span> arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional <em>weight</em> vector and a positive <em>profit</em>, along with <em>m</em> 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.</p><p>Our main result is a <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ln</mi><mo></mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for 2VMK, for every fixed <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, thus improving the best known ratio of <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mo>−</mo><mi>ε</mi><mo>)</mo></math></span> which follows as a special case from a result of Fleischer et al. (2011) <span><span>[6]</span></span>.</p><p>Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) <span><span>[15]</span></span>, originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to <span><math><mo>≈</mo><mi>m</mi><mo>⋅</mo><mi>ln</mi><mo></mo><mn>2</mn><mo>≈</mo><mn>0.693</mn><mo>⋅</mo><mi>m</mi></math></span> of the bins, followed by a reduction to the (1-dimensional) <span>Multiple Knapsack</span> problem for assigning items to the remaining bins.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"124 ","pages":"Article 102124"},"PeriodicalIF":0.4,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0925772124000464/pdfft?md5=aabf82f5f8cf463934bfaf0d08024ae5&pid=1-s2.0-S0925772124000464-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-22DOI: 10.1016/j.comgeo.2024.102122
Gang Liu, Haitao Wang
Given a set P of n points and a set S of m disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line ℓ. We present an time algorithm for the problem. This improves the previously best result of time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of S are located on a line ℓ while points of P can be anywhere in the plane. Our algorithm runs in time, which improves the previously best result of time. In addition, our results lead to an algorithm of time for a half-plane coverage problem (given n half-planes and n points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of time. Further, if all half-planes are lower ones, our algorithm runs in time while the previously best algorithm takes time.
{"title":"On the line-separable unit-disk coverage and related problems","authors":"Gang Liu, Haitao Wang","doi":"10.1016/j.comgeo.2024.102122","DOIUrl":"10.1016/j.comgeo.2024.102122","url":null,"abstract":"<div><p>Given a set <em>P</em> of <em>n</em> points and a set <em>S</em> of <em>m</em> disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of <em>P</em>. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of <em>P</em> by a line <em>ℓ</em>. We present an <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mo>)</mo></math></span> time algorithm for the problem. This improves the previously best result of <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>m</mi><mo>+</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of <em>S</em> are located on a line <em>ℓ</em> while points of <em>P</em> can be anywhere in the plane. Our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>m</mi><mi>log</mi><mo></mo><mi>m</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time, which improves the previously best result of <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>m</mi><mi>log</mi><mo></mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time. In addition, our results lead to an algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time for a half-plane coverage problem (given <em>n</em> half-planes and <em>n</em> points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time. Further, if all half-planes are lower ones, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time while the previously best algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"123 ","pages":"Article 102122"},"PeriodicalIF":0.4,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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