On additive spanners in weighted graphs with local error

An \emph{additive $+\beta$ spanner} of a graph $G$ is a subgraph which preserves distances up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al.\ 2019 and 2020, Ahmed et al.\ 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al.\ 2020] provided constructions of sparse spanners with \emph{global} error $\beta =cW$, where $W$ is the maximum edge weight in $G$ and $c$ is constant. We improve these to \emph{local} error by giving spanners with additive error $+cW(s,t)$ for each vertex pair $(s,t)$, where $W(s,t)$ is the maximum edge weight along the shortest $s$–$t$ path in $G$. These include pairwise $+(2+\text{eps})W(\cdot ,\cdot )$ and $+(6+\eps) W(\cdot, \cdot)$spannersoververtexpairs$\Pc \subseteq V \times V$on$O_{\eps}(n|\Pc|^{1/3})$and$O_{\eps}(n|\Pc|^{1/4})$edgesforall$\eps > 0$, which extend previously known unweighted results up to $\text{eps}$ dependence, as well as an all-pairs $+4W(\cdot ,\cdot )$ spanner on $\tilde{O}(n^{7\mathrm{/}5})$ edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its \emph{lightness}, defined as the total edge weight of the spanner divided by the weight of an MST of $G$. We provide a $+\eps W(\cdot,\cdot)$spannerwith$O_{\eps}(n)$lightness,anda$+(4+\eps) W(\cdot,\cdot)$spannerwith$O_{\eps}(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.