Related papers: Lightweight Near-Additive Spanners
Let $G$ be an unweighted, undirected graph. An additive $k$-spanner of $G$ is a subgraph $H$ that approximates all distances between pairs of nodes up to an additive error of $+k$, that is, it satisfies $d_H(u,v) \le d_G(u,v)+k$ for all…
For a weighted graph $G = (V, E, w)$ and a designated source vertex $s \in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT).…
Given a set $S$ of $n$ points in the plane and a parameter $\varepsilon>0$, a Euclidean $(1+\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that contains, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)\|pq\|$. We…
Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners…
We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni,…
The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+\epsilon)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(\epsilon^{-d})$ for any $d =…
Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A…
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…
In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+\epsilon)$-spanner of…
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a…
Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say $\tilde{O}(n^{1+\delta})$ bits. There is an inherent tradeoff between the…
For $\alpha \ge 1$, $\beta \ge 0$, and a graph $G$, a spanning subgraph $H$ of $G$ is said to be an $(\alpha, \beta)$-spanner if $\dist(u, v, H) \le \alpha \cdot \dist(u, v, G) + \beta$ holds for any pair of vertices $u$ and $v$. These type…
Lightness, sparsity, and hop-diameter are the fundamental parameters of geometric spanners. Arya et al. [STOC'95] showed in their seminal work that there exists a construction of Euclidean $(1+\varepsilon)$-spanners with hop-diameter…
We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work…
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$,…
A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as…
Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have $(1+\epsilon)$-spanners that are light, that is, of weight $g(|H|,\epsilon)$ times the weight of the minimum spanning tree for some function $g$. This conjecture implies…
Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in…
In a directed graph $G$ with non-correlated edge lengths and costs, the \emph{network design problem with bounded distances} asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria…
We construct $n$-node graphs on which any $O(n)$-size spanner has additive error at least $+\Omega(n^{3/17})$, improving on the previous best lower bound of $\Omega(n^{1/7})$ [Bodwin-Hoppenworth FOCS '22]. Our construction completes the…