Approximation Algorithms for Directed Weighted Spanners
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In the pairwise weighted spanner problem, the input consists of an n-vertex-directed graph, where each edge is assigned a cost and a length. Given k vertex pairs and a distance constraint for each pair, the goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. This formulation captures many well-studied connectivity problems, including spanners, distance preservers, and Steiner forests. In the offline setting, we show: 1. An Õ(n^4/5 + ϵ)-approximation algorithm for pairwise weighted spanners. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n^3/5 + ϵ)-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (TALG, 2020). 2. An Õ(n^1/2+ϵ)-approximation algorithm for all-pair weighted distance preservers. When the edges have unit costs and arbitrary lengths, the best previous algorithm gives an Õ(n^1/2)-approximation for all-pair spanners, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). In the online setting, we show: 1. An Õ(k^1/2 + ϵ)-competitive algorithm for pairwise weighted spanners. The state-of-the-art results are Õ(n^4/5)-competitive when edges have unit costs and arbitrary lengths, and min{Õ(k^1/2 + ϵ), Õ(n^2/3 + ϵ)}-competitive when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021). 2. An Õ(k^ϵ)-competitive algorithm for single-source weighted spanners. Without distance constraints, this problem is equivalent to the directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is Õ(k^ϵ)-competitive, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018).
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