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 $\tilde{O}(n^{4/5 + \epsilon})$-approximation algorithm for pairwise weighted spanners. When the edges have unit costs and lengths, the best previous algorithm gives an $\tilde{O}(n^{3/5 + \epsilon})$-approximation, due to Chlamt\'a\v{c}, Dinitz, Kortsarz, and Laekhanukit (TALG, 2020). 2. An $\tilde{O}(n^{1/2+\epsilon})$-approximation algorithm for all-pair weighted distance preservers. When the edges have unit costs and arbitrary lengths, the best previous algorithm gives an $\tilde{O}(n^{1/2})$-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 $\tilde{O}(k^{1/2 + \epsilon})$-competitive algorithm for pairwise weighted spanners. The state-of-the-art results are $\tilde{O}(n^{4/5})$-competitive when edges have unit costs and arbitrary lengths, and $\min\{\tilde{O}(k^{1/2 + \epsilon}), \tilde{O}(n^{2/3 + \epsilon})\}$-competitive when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021). 2. An $\tilde{O}(k^{\epsilon})$-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 $\tilde{O}(k^{\epsilon})$-competitive, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018).

翻译：在成对加权支撑树问题中，输入包含一个$n$个顶点的有向图，其中每条边被赋予代价和长度。给定$k$个顶点对及每对顶点的距离约束，目标是找到一个最小代价子图，使得所有距离约束得到满足。该形式化描述涵盖了许多被广泛研究的连通性问题，包括支撑树、距离保持器和斯坦纳森林。在离线场景下，我们证明：1. 对于成对加权支撑树，存在一个$\tilde{O}(n^{4/5 + \epsilon})$-近似算法。当边具有单位代价和长度时，先前最优算法由Chlamtáč、Dinitz、Kortsarz和Laekhanukit（TALG, 2020）给出，其近似比为$\tilde{O}(n^{3/5 + \epsilon})$。2. 对于全对加权距离保持器，存在一个$\tilde{O}(n^{1/2+\epsilon})$-近似算法。当边具有单位代价和任意长度时，针对全对支撑树的先前最优算法由Berman、Bhattacharyya、Makarychev、Raskhodnikova和Yaroslavtsev（Information and Computation, 2013）给出，其近似比为$\tilde{O}(n^{1/2})$。在线场景下，我们证明：1. 对于成对加权支撑树，存在一个$\tilde{O}(k^{1/2 + \epsilon})$-竞争算法。当前最优结果为：当边具有单位代价和任意长度时，竞争比为$\tilde{O}(n^{4/5})$；当边具有单位代价和长度时，竞争比为$\min\{\tilde{O}(k^{1/2 + \epsilon}), \tilde{O}(n^{2/3 + \epsilon})\}$，由Grigorescu、Lin和Quanrud（APPROX, 2021）提出。2. 对于单源加权支撑树，存在一个$\tilde{O}(k^{\epsilon})$-竞争算法。当无距离约束时，该问题等价于有向斯坦纳树问题。在线有向斯坦纳树的先前最优算法由Chakrabarty、Ene、Krishnaswamy和Panigrahi（SICOMP, 2018）提出，其竞争比为$\tilde{O}(k^{\epsilon})$。