Related papers: Time-dependent shortest paths in bounded treewidth…

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph $G$ with a linear arrangement of bandwidth $b$ can be embedded into a distribution…

Let $P$ be a set of $n$ points in the plane where each point $p$ of $P$ is associated with a radius $r_p>0$.The transmission graph $G=(P,E)$ of $P$ is defined as the directed graph such that $E$ contains an edge from $p$ to $q$ if and only…

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result…

The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e. For unweighted directed graphs on n vertices, the…

We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This…

We present an algorithm that computes the girth of the intersection graph of $n$ given line segments in the plane in $O(n^{1.483})$ expected time. This is the first such algorithm with $O(n^{3/2-\varepsilon})$ running time for a positive…

The expected meeting time of two random walkers on an undirected graph of size $N$, where at each time step one walker moves and the process stops when they collide, satisfies a system of $\binom{N}{2}$ linear equations. Na\"{i}vely,…

In the \emph{$k$-Diameter-Optimally Augmenting Tree Problem} we are given a tree $T$ of $n$ vertices as input. The tree is embedded in an unknown \emph{metric} space and we have unlimited access to an oracle that, given two distinct…

We give an algorithm that takes as input an $n$-vertex graph $G$ and an integer $k$, runs in time $2^{O(k^2)} n^{O(1)}$, and outputs a tree decomposition of $G$ of width at most $k$, if such a decomposition exists. This resolves the…

Given an $n$-vertex planar directed graph with real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in $O(n\log^2n/\log\log n)$ time with O(n) space. This is an improvement…

We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise…

In this note we describe an application of low-high orders in fault-tolerant network design. Baswana et al. [DISC 2015] study the following reachability problem. We are given a flow graph $G = (V, A)$ with start vertex $s$, and a spanning…

We give an algorithm that for an input n-vertex graph G and integer k>0, in time 2^[O(k)]n either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k+4. This is the first algorithm…

We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in…

Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(gn log n) time…

In this paper, we present efficient algorithms for the single-source shortest path problem in weighted disk graphs. A disk graph is the intersection graph of a family of disks in the plane. Here, the weight of an edge is defined as the…

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M…

Enumerating the minimal hitting sets of a hypergraph is a problem which arises in many data management applications that include constraint mining, discovering unique column combinations, and enumerating database repairs. Previously, Eiter…

Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with $o(mn)$ total update time, where $m$ is the number of edges and $n$ is the…

Minimum spanning trees are important tools in the analysis and design of networks. Many practical applications require their computation, ranging from biology and linguistics to economy and telecommunications. The set of cycles of a network…