Stable Approximation Algorithms for the Dynamic Broadcast Range-Assignment Problem
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Let P be a set of points in ℝ^d, where each point p∈ P has an associated transmission range ρ(p). The range assignment ρ induces a directed communication graph 𝒢_ρ(P) on P, which contains an edge (p,q) iff |pq| ≤ρ(p). In the broadcast range-assignment problem, the goal is to assign the ranges such that 𝒢_ρ(P) contains an arborescence rooted at a designated node and whose cost ∑_p ∈ Pρ(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 ε>0, is k(ϵ)-stable and maintains a solution with approximation ratio 1+ε, where the stability parameter k(ε) only depends on ε and not on the size of P. We study such trade-offs in three settings. - In ℝ^1, we present a SAS with k(ε)=O(1/ε), which we show is tight in the worst case. We also present a 1-stable (6+2√(5))-approximation algorithm, a 2-stable 2-approximation algorithm, and a 3-stable 1.97-approximation algorithm. - In 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 ℝ^1. - In ℝ^2, we also prove that no SAS exists, and we present a O(1)-stable O(1)-approximation algorithm.
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