Related papers: Approximate Spanning Tree Counting from Uncorrelat…
Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be…
We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded…
We present a linear programming based algorithm for computing a spanning tree $T$ of a set $P$ of $n$ points in $\Re^d$, such that its crossing number is $O(\min(t \log n, n^{1-1/d}))$, where $t$ the minimum crossing number of any spanning…
We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with $ n $ nodes and $ m $ edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in…
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…
Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems…
In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and…
We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the…
We study the problem of estimating the number of edges of an unknown, undirected graph $G=([n],E)$ with access to an independent set oracle. When queried about a subset $S\subseteq [n]$ of vertices the independent set oracle answers whether…
We give the first almost-linear time algorithm for computing the \emph{maximal $k$-edge-connected subgraphs} of an undirected unweighted graph for any constant $k$. More specifically, given an $n$-vertex $m$-edge graph $G=(V,E)$ and a…
The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an $n$-node input graph to be read sequentially in $p$ passes using $\tilde{O}(n)$ space. In this…
We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight…
We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}}…
We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input $n$-vertex $m$-edge hypergraph $H$ of fractional hypertree width at most $\omega$, runs in…
Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected…
Given an undirected, weighted graph, the minimum spanning tree (MST) is a tree that connects all of the vertices of the graph with minimum sum of edge weights. In real world applications, network designers often seek to quickly find a…
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree…
Given an $n$-vertex $m$-edge graph $G$ with non negative edge-weights, the girth of $G$ is the weight of a shortest cycle in $G$. For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that…
We study a natural problem in graph sparsification, the Spanning Tree Congestion (\STC) problem. Informally, the \STC problem seeks a spanning tree with no tree-edge \emph{routing} too many of the original edges. The root of this problem…
We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate…