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 $+(2+eps)W(cdot,cdot)$ and $+(6+eps) W(cdot, cdot)$ spanners over vertex pairs $Pc subseteq V times V$ on $O_{eps}(n|Pc|^{1/3})$ and $O_{eps}(n|Pc|^{1/4})$ edges for all $eps > 0$, which extend previously known unweighted results up to $eps$ dependence, as well as an all-pairs $+4W(cdot,cdot)$ spanner on $widetilde{O}(n^{7/5})$ 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 $+eps W(cdot,cdot)$ spanner with $O_{eps}(n)$ lightness, and a $+(4+eps) W(cdot,cdot)$ spanner with $O_{eps}(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.