Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-02-09 DOI:10.1137/23m1545975

Mark de Berg, Arpan Sadhukhan, Frits Spieksma

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 790-827, March 2024.
Abstract. Let [math] be a set of points in [math], where each point [math] has an associated transmission range, denoted [math]. The range assignment [math] induces a directed communication graph [math] on [math], which contains an edge [math] iff [math]. In the broadcast range-assignment problem, the goal is to assign the ranges such that [math] contains an arborescence rooted at a designated root node and the cost [math] of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution—the number of ranges that are modified when a point is inserted into or deleted from [math]—and its approximation ratio. To this end we study [math]-stable algorithms, which are algorithms that modify the range of at most [math] points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm [math] that, for any given fixed parameter [math], is [math]-stable and that maintains a solution with approximation ratio [math], where the stability parameter [math] only depends on [math] and not on the size of [math]. We study such trade-offs in three settings. (1) For the problem in [math], we present a SAS with [math]. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have [math]. We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in [math] (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in [math], we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in [math]. (3) For the problem in [math], we also prove that no SAS exists, and we present a [math]-stable [math]-approximation algorithm. Most results generalize to the setting where, for any given constant [math], the range-assignment cost is [math].

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动态广播范围分配问题的稳定近似算法

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 790-827 页，2024 年 3 月。摘要。设[math]是[math]中的一组点，其中每个点[math]都有一个相关的传输范围，记为[math]。范围分配[math]在[math]上引起一个有向通信图[math]，如果[math]包含一条边[math]。在广播范围分配问题中，我们的目标是分配范围，使[math]包含一个以指定根节点为根的树状图，且分配的代价[math]最小。我们研究这个问题的动态版本。特别是，我们研究了解的稳定性--当[math]中插入或删除一个点时修改的范围数--与其近似率之间的权衡。为此，我们研究了[math]稳定算法，即在更新解时最多只修改[math]个点的范围的算法。我们还引入了稳定近似方案（简称 SAS）的概念。SAS 是一种更新算法[math]，对于任何给定的固定参数[math]，它都是[math]稳定的，并且能保持一个近似率为[math]的解，其中稳定参数[math]只取决于[math]，而不取决于[math]的大小。我们在三种情况下研究这种权衡。(1) 对于[math]中的问题，我们提出了一个具有[math]的 SAS。此外，我们还证明，在最坏的情况下这是紧密的：问题的任何 SAS 都必须有 [math]。我们还提出了具有恒定近似率的 1-、2- 和 3-稳定算法。(2) 对于 [math] 中的问题（即当底层空间是圆时），我们证明不存在任何 SAS。尽管对于[math]中的静态问题，我们证明了总是可以通过在适当的点上切割圆并求解[math]中的问题来获得最优解。(3) 对于[math]中的问题，我们也证明不存在 SAS，并提出了一种[math]稳定的[math]逼近算法。对于任何给定常数[math]，范围分配成本为[math]，大多数结果都可以推广到这种情况。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.