Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem

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].