CS 269S, Spring 2021: Theory Seminar

May 21, 2021, 1:00 – 1:50:

On Additive Spanners in Weighted Graphs with Local Error

Hadi Khodabandeh

An 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 global error $\beta =cW$, where $W$ is the maximum edge weight in $G$ and $c$ is constant. We improve these to 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 $+(2+\epsilon )W(\cdot ,\cdot )$ and $+(6+\epsilon )W(\cdot ,\cdot )$ spanners over vertex pairs $\mathcal{P}\subseteq V\times V$ on $O_{\epsilon }(n|\mathcal{P}{|}^{1/3})$ and $O_{\epsilon }(n|\mathcal{P}{|}^{1/4})$ edges for all $\epsilon >0$, which extend previously known unweighted results up to $\epsilon$ dependence, as well as an all-pairs $+4W(\cdot ,\cdot )$ spanner on $\tilde{O}(n^{7/5})$ edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of a minimum spanning tree of $G$. We provide a $+\epsilon W(\cdot ,\cdot )$ spanner with $O_{\epsilon }(n)$ lightness, and a $+(4+\epsilon )W(\cdot ,\cdot )$ spanner with $O_{\epsilon }(n^{2/3})$ lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

Based on a paper by Reyan Ahmed, Greg Bodwin, Keaton Hamm, Stephen Kobourov, and Richard Spence, arXiv:2103.09731.