相关论文: On the Integrality Gap of the Directed-Component R…
Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a 1.39-approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a…
The Steiner Tree problem asks for the cheapest way of connecting a given subset of the vertices in an undirected graph. One of the most prominent linear programming relaxations for Steiner Tree is the Bidirected Cut Relaxation (BCR).…
A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least $36/31$, and it has long been conjectured that its true…
We consider the Steiner tree problem in quasi-bipartite graphs, where no two Steiner vertices are connected by an edge. For this class of instances, we present an efficient algorithm to exactly solve the so called directed component…
The bidirected cut relaxation is the characteristic representative of the bidirected relaxations ($\mathrm{\mathcal{BCR}}$) which are a well-known class of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs (STP).…
We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is $O(\log k)$ in quasi-bipartite graphs with $k$ terminals. Such instances can be seen to generalize set cover, so the…
The Steiner tree problem is one of the most prominent problems in network design. Given an edge-weighted undirected graph and a subset of the vertices, called terminals, the task is to compute a minimum-weight tree containing all terminals…
The Directed Steiner Tree (DST) problem is defined on a directed graph $G=(V,E)$, where we are given a designated root vertex $r$ and a set of $k$ terminals $K \subseteq V \setminus {r}$. The goal is to find a minimum-cost subgraph that…
We study the Directed Steiner Tree (DST) problem in layered graphs through a simple path-based linear programming relaxation. This relaxation achieves an integrality gap of O(l log k), where k is the number of terminals and l is the number…
In the Directed Steiner Tree (DST) problem, we are given a directed graph $G=(V,E)$ on $n$ vertices with edge-costs $c \in \mathbb{R}_{\geq 0}^E$, a root vertex $r \in V$, and a set $K \subseteq V \setminus \{r\}$ of $k$ terminals. The goal…
We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the ODD-LP relaxation for the (weighted) Tree Augmentation Problem for a $k$-level tree instance is at most $2 - \frac{1}{2^{k-1}}$.…
The Steiner Forest problem is an important generalization of the Steiner Tree problem. We are given an undirected graph with nonnegative edge costs and a collection of pairs of vertices. The task is to compute a cheapest forest with the…
Until recently, LP relaxations have played a limited role in the design of approximation algorithms for the Steiner tree problem. In 2010, Byrka et al. presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation, but…
The Steiner tree problem is one of the classic and most fundamental $\mathcal{NP}$-hard problems: given an arbitrary weighted graph, seek a minimum-cost tree spanning a given subset of the vertices (terminals). Byrka \emph{et al}. proposed…
In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (BCR) formulation and introducing a novel formulation, the Complete Metric (CM) model,…
We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the…
Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random…
The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G=(V,E), a set of terminals R, and non-negative costs c_e for…
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs $\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{\pi_i\}_{i=1}^k$ for each terminal pair; the…
The goal for the Directed Steiner Tree problem is to find a minimum cost tree in a directed graph G=(V,E) that connects all terminals X to a given root r. It is well known that modulo a logarithmic factor it suffices to consider acyclic…