Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem

Let $P$ be a set of points in $\mathbb{R}^d$, where each point $p\in P$ has an associated transmission range $\rho(p)$. The range assignment $\rho$ induces a directed communication graph $\mathcal{G}_{\rho}(P)$ on $P$, which contains an edge $(p,q)$ iff $|pq| \leq \rho(p)$. In the broadcast range-assignment problem, the goal is to assign the ranges such that $\mathcal{G}_{\rho}(P)$ contains an arborescence rooted at a designated node and whose cost $\sum_{p \in P} \rho(p)^2$ is minimized. 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 $P$ -- and its approximation ratio. We introduce $k$-stable algorithms, which are algorithms that modify the range of at most $k$ points when they update the solution. We also introduce the concept of a stable approximation scheme (SAS). A SAS is an update algorithm that, for any given fixed parameter $\varepsilon>0$, is $k(\epsilon)$-stable and maintains a solution with approximation ratio $1+\varepsilon$, where the stability parameter $k(\varepsilon)$ only depends on $\varepsilon$ and not on the size of $P$. We study such trade-offs in three settings. - In $\mathbb{R}^1$, we present a SAS with $k(\varepsilon)=O(1/\varepsilon)$, which we show is tight in the worst case. We also present a 1-stable $(6+2\sqrt{5})$-approximation algorithm, a $2$-stable 2-approximation algorithm, and a $3$-stable $1.97$-approximation algorithm. - In $\mathbb{S}^1$ (where the underlying space is a circle) we prove that no SAS exists, even though an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in $\mathbb{R}^1$. - In $\mathbb{R}^2$, we also prove that no SAS exists, and we present a $O(1)$-stable $O(1)$-approximation algorithm.