Related papers: Additive One Approximation for Minimum Degree Span…
Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with…
We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…
In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the \emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods…
In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a…
In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied…
Network interdiction problems are a natural way to study the sensitivity of a network optimization problem with respect to the removal of a limited set of edges or vertices. One of the oldest and best-studied interdiction problems is…
We show an $\widetilde{O}(m^{1.5} \epsilon^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+\epsilon)$ factor with high probability, improving on the previous…
A fundamental problem in wireless networks is the \emph{minimum spanning tree} (MST) problem: given a set $V$ of wireless nodes, compute a spanning tree $T$, so that the total cost of $T$ is minimized. In recent years, there has been a lot…
We consider the problem of enumerating, for a given directed graph $G=(V,E)$ and a node $r\in V$, all directed spanning trees of $G$ rooted at $r$. For undirected graphs, the corresponding problem of enumerating all spanning trees has…
We study budget constrained network upgradeable problems. We are given an undirected edge weighted graph $G=(V,E)$ where the weight an edge $e \in E$ can be upgraded for a cost $c(e)$. Given a budget $B$ for improvement, the goal is to find…
A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…
For an $m$-edge connected simple graph $G$, finding a spanning tree of $G$ with the maximum number of leaves is MAXSNP-complete. The problem remains NP-complete even if $G$ is planar and the maximal degree of $G$ is at most four. Lu and…
With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The…
We investigate the time-complexity of the All-Pairs Max-Flow problem: Given a graph with $n$ nodes and $m$ edges, compute for all pairs of nodes the maximum-flow value between them. If Max-Flow (the version with a given source-sink pair…
The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…
Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq…
Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the…
The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques…
This paper focuses on finding a spanning tree of a graph to maximize the number of its internal vertices. We present an approximation algorithm for this problem which can achieve a performance ratio $\frac{4}{3}$ on undirected simple…
The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation…