首页/论文/0908.3652

arXiv 0908.36522009-08-25cs.DScs.CG

最优几何划分、覆盖与K-中心

Optimal Geometric Partitions, Covers and K-Centers

Mugurel Ionut Andreica, Eliana-Dina Tirsa, Cristina Teodora Andreica, Romulus Andreica, Mihai Aristotel Ungureanu

备注：: Extended version of the conference paper
发表信息：: Proc. of the 9th WSEAS Intl. Conf. on Mathematics and Computers in Business and Economics (MCBE), pp. 173-178, Bucharest, Romania, 24-26 June, 2008. (ISBN: 978-960-6766-76-3 / ISSN: 1790-5109)

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

摘要：

在本文中，我们提出了一些新的，实用的，几何优化技术计算多边形分区，1D和2D点，间隔，正方形和矩形覆盖，以及1D和2D的间隔和矩形K-中心。我们提出的所有技术都可以直接应用于实际中非常常见的几个成本优化和设施选址问题。所采用的主要技术是动态规划，但我们也利用有效的数据结构和快速贪心算法。

英文摘要：

In this paper we present some new, practical, geometric optimization techniques for computing polygon partitions, 1D and 2D point, interval, square and rectangle covers, as well as 1D and 2D interval and rectangle K-centers. All the techniques we present have immediate applications to several cost optimization and facility location problems which are quite common in practice. The main technique employed is dynamic programming, but we also make use of efficient data structures and fast greedy algorithms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Minmax Centered k-Partitioning of Trees and Applications to Sink Evacuation with Dynamic Confluent Flows

Let $T=(V,E)$ be a tree with associated costs on its subtrees. A minmax $k$-partition of $T$ is a partition into $k$ subtrees, minimizing the maximum cost of a subtree over all possible partitions. In the centered version of the problem, the cost of a subtree cost is defined as the minimum cost of "servicing" that subtree using a center located within it. The problem motivating this work was the sink-evacuation problem on trees, i.e., finding a collection of $k$-sinks that minimize the time required by a confluent dynamic network flow to evacuate all supplies to sinks. This paper provides the …

2018-03-25cs.DS

边分块顶点覆盖问题的近似算法

Approximation Algorithms for Edge Partitioned Vertex Cover Problems

我们考虑部分顶点覆盖问题的自然推广。这里，实例由图G =（V，E）、正成本函数c：V-> Z^{+}、边集$E$的分区$P_1,..., P_r$以及每个分区$P_i$的参数$k_i$组成。目标是找到覆盖来自分区$P_i$的至少$k_i$边的顶点的最小成本集合。我们称之为分区顶点覆盖问题。本文给出了该问题逼近性的匹配上下界。我们的算法是基于一个新的LP松弛这个问题。这个LP松弛通过将背包覆盖不等式添加到问题的自然LP松弛来获得。我们证明了这个线性规划的完整性间隙为$O(log r)$，其中$r$是边集划分中的集合数。我们还将我们的结果扩展到更一般的设置。

2012-10-10cs.DS

带容量的划分顶点覆盖和划分边覆盖

Capacitated Partition Vertex Cover and Partition Edge Cover

Northwestern University（西北大学）

研究提出带容量的划分顶点覆盖和划分边覆盖问题，设计了两种近似算法并改进了边覆盖问题的算法效率。

2026-09-28cs.DS

On the Partition Set Cover Problem

Several algorithms with an approximation guarantee of $O(\log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ \emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $\mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(β+ \log r)$ approximation for the Partition Set Cover problem. Here $β$ denotes the approximation guarantee for a relat…

2018-12-01cs.DS

具有划分与公平性约束的子模覆盖的双标准算法

Bicriteria Algorithms for Submodular Cover with Partition and Fairness Constraints

Texas A&M University（德克萨斯农工大学）

本文提出双标准近似算法，解决具有划分与公平性约束的子模覆盖问题，实现最优近似保证并提升算法效率。

2026-01-16cs.DS

从分层划分到分层覆盖：二重映射的最优容错生成器

From Hierarchical Partitions to Hierarchical Covers: Optimal Fault-Tolerant Spanners for Doubling Metrics

在本文中，我们设计了一个最佳结构的容错空间加倍度量。具体地说，对于任意$n$ -点加倍度量、任意$\eps > 0$和任意整数$0 \le k \le n-2$，我们的构造提供了一个在最优时间$O(n \log n + k n)$内具有最优度$O(k)$的$k$ -容错$(1+\eps)$ -模型.然后，我们加强这一结果，以提供接近最佳的（$\log k$的一个因素）保证我们的spectum的直径和重量，即直径$O(\log n)$和重量$O(k^2 + k \log n) \cdot ω(MST)$，同时保持最佳的保证程度$O(k)$和运行时间$O(n \log n + k n)$。我们的结果解决了这一领域的几个基本的开放问题，最终结束了从Arya等人的STOC'95论文开始的一系列研究。以及Levcopoulos等人的STOC'98论文。在得到这一结果的过程中，我们发展了一种新的技术，用于在加倍度量中构造空间。我们的构造是基于一个新的\emph{hierarchical cover}的度量，而大多数以前的构造spiritual加倍和欧几里德度量（如网树的spiritual）是基于\emph{hierarchical partitions}的度量。我们证明了权力的层次覆盖的背景下，几何speries通过提高国家的最先进的结果在这方面。

2013-05-08cs.CGcs.DS

更多相关论文收起更多论文

加权划分顶点覆盖与边覆盖

Weighted Partition Vertex and Edge Cover

本文研究带分组覆盖约束的加权顶点覆盖与边覆盖问题，为WP-PVC提出2-近似和双准则算法，为W-PEC给出更快的精确多项式时间算法，并证明其奖品收集变体NP完全。

2025-08-18cs.DS

A local search $4/3$-approximation algorithm for the minimum $3$-path partition problem

Given a graph $G = (V, E)$, the $3$-path partition problem is to find a minimum collection of vertex-disjoint paths each of order at most $3$ to cover all the vertices of $V$. It is different from but closely related to the well-known $3$-set cover problem. The best known approximation algorithm for the $3$-path partition problem was proposed recently and has a ratio $13/9$. Here we present a local search algorithm and show, by an amortized analysis, that it is a $4/3$-approximation. This ratio matches up to the best approximation ratio for the $3$-set cover problem.

2018-12-21cs.DS

具有一般目标函数的集合覆盖问题

Set Covering Problems with General Objective Functions

我们引入了一个参数化版本的集合覆盖，概括了以前研究的几个问题。给定基集V和V的子集集合S_i，可行解是V的划分，使得划分的每个子集包含在S_i之一中。这个问题涉及到最大化分区的平均子集大小，其中平均值是参数p的广义平均值，取所有元素。当p=-1时，该问题等价于经典的最小集覆盖问题.对于p=0，它等价于Halperin和Karp提出的最小熵集合覆盖问题。对于p=1，该问题包括最大边团划分问题作为特殊情况。我们证明了对于R^+中的任意p，贪心算法同时在（p+1）^1/p的因子内逼近问题，并且这是最好的可能，除非P=NP。这些结果推广和简化了以前的特殊情况下的结果。我们还考虑了相应的图着色问题，证明了几个易处理性和不可逼近性结果。最后，我们考虑进一步推广的集合覆盖问题，我们的目标是最小化的部分大小的一些凹函数的总和。作为一个应用，我们得到了一个近似比租或买集覆盖问题。

2008-02-15cs.DS

无小图中的捷径划分：Steiner点移除、距离预言机、树覆盖及更多应用

Shortcut Partitions in Minor-Free Graphs: Steiner Point Removal, Distance Oracles, Tree Covers, and More

本文突破了平面性限制，通过引入cop分解的确定性变体，首次为$K_r$-minor-free图构建了捷径划分，并据此实现了最优距离预言机和常数大小的树覆盖。

2023-07-31cs.DS

树和有界树宽图上路径覆盖与路径划分的多项式时间算法

Polynomial-time algorithms for PATH COVER and PATH PARTITION on trees and graphs of bounded treewidth

本文针对研究较少的 PATH COVER 问题，提出树上线性时间算法及有界树宽图上的 XP 动态规划算法，并给出 PATH PARTITION 的 FPT 算法，结果可推广到诱导路径和边不相交变体。

2025-11-10cs.DMcs.DS

独立集之上的边团划分与覆盖

Edge Clique Partition and Cover Beyond Independence

本文研究了边团覆盖和划分问题高于最大独立集α的参数化，证明了ECP/α固定参数可解，而ECC/α在k≥2时NP完全，并在以k+ω(G)为参数时变得固定参数可解。

2025-06-26cs.DMcs.DS

A Technique for Obtaining True Approximations for $k$-Center with Covering Constraints

There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the $k$-Center problem in this spirit are Colorful $k$-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust $k$-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional $k$-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard const…

2020-07-08cs.DSmath.OC

The Solidarity Cover Problem

Various real-world problems consist of partitioning a set of locations into disjoint subsets, each subset spread in a way that it covers the whole set with a certain radius. Given a finite set S, a metric d, and a radius r, define a subset (of S) S' to be an r-cover if and only if forall s in S there exists s' in S' such that d(s,s') is less or equal to r. We examine the problem of determining whether there exist m disjoint r-covers, naming it the Solidarity Cover Problem (SCP). We consider as well the related optimization problems of maximizing the number of r-covers, referred to as the parti…

2023-02-07cs.CCcs.CGcs.DS

Approximation algorithms for the directed path partition problems

Given a directed graph $G = (V, E)$, the $k$-path partition problem is to find a minimum collection of vertex-disjoint directed paths each of order at most $k$ to cover all the vertices of $V$. The problem has various applications in facility location, network monitoring, transportation and others. Its special case on undirected graphs has received much attention recently, but the general directed version is seemingly untouched in the literature. We present the first $k/2$-approximation algorithm, for any $k \ge 3$, based on a novel concept of augmenting path to minimize the number of singleto…

2021-07-09cs.DS

无线传感器网络能量有效监测的Set K覆盖算法

Set K-Cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks

无线传感器网络（WSNs）是环境监测的有效手段。本文研究了一种在无线传感器网络的能源有效的监测策略，划分成覆盖的传感器，然后激活覆盖迭代循环的方式。这种方法利用了当许多传感器监视单个区域时产生的重叠。我们的工作建立在Slijepcevic和Potterak的“无线传感器网络的功率高效组织”中的先前工作的基础上，该模型首次制定。我们已经设计了三个近似算法的一个变化的设置K-覆盖问题，其目标是分区的传感器到覆盖，使覆盖的数量，包括一个区域，在所有领域的总和，是最大化的。第一种算法是随机化的，并将传感器分区，预期在最佳值的分数1 - 1/e（~.63）内。我们提出了另外两个确定性的近似算法。一种是近似比为1/2的分布式贪心算法，另一种是近似比为1 - 1/e的集中式贪心算法。我们表明，它是NP-完全的，以保证优于15/16的最佳覆盖率，这表明所有三种算法的最佳近似算法的性能。模拟表明，在实践中，确定性算法执行远高于其最坏情况下的界限，始终覆盖超过72%的最佳解决方案所覆盖的。模拟还表明，寿命的增加与传感器之间的重叠量成比例。该算法速度快，易于使用，并根据模拟，显着增加传感器网络的寿命。特别是随机算法似乎相当实用。

2003-11-20cs.DS

A Constant-Factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a …

2020-09-15cs.DS

用于背包约束聚类和分区系统的依赖随机舍入

Dependent randomized rounding for clustering and partition systems with knapsack constraints

本文研究带背包与分区约束的公平聚类，提出依赖随机舍入算法，为多背包中位数和中心问题给出新的近似与伪近似保证，并建立无界方差随机变量和的新尾界。

2024-05-11cs.DS

一种面向分词的分区覆盖方法

A Partition Cover Approach to Tokenization

Singapore Management University（新加坡管理大学）；MIT-IBM Watson AI Lab（MIT-IBM沃森人工智能实验室）；Harvard University（哈佛大学）；National University of Singapore（新加坡国立大学）

本文将分词建模为优化问题并证明其 NP-hard，提出多项式贪心算法 GreedTok，实验表明其压缩效果优于 BPE 和 Unigram，并可提升十亿参数语言模型的每字节比特效率。

2025-09-28cs.AIcs.CLcs.DS