Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

Efficient Algorithms for Solving Hypergraphic Steiner Tree Relaxations in Quasi-Bipartite Instances

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 relaxation (DCR), a specific form of hypergraphic LP relaxation that was instrumental in the recent break-through result by Byrka et al. [BGRS10] (STOC 2010). Our algorithm hinges on an efficiently computable map from extreme points of the bidirected cut relaxation to feasible solutions of (DCR). As a consequence, together with [BGRS10] we immediately obtain an eff…

Matroids and Integrality Gaps for Hypergraphic Steiner Tree Relaxations

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 surprisingly, their analysis does not provide a matching bound on the integrality gap. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem - one that heavily exploits methods and results from the theory of matroids and submodular functions - which leads to stronger integrality gaps, faster algorithms, and a variety of structural insig…

Hypergraphic LP Relaxations for Steiner Trees

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 integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number o…

Simplex based Steiner tree instances yield large integrality gaps for the bidirected cut relaxation

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). Although no general approximation algorithm based on $\mathrm{\mathcal{BCR}}$ with an approximation ratio better than $2$ for STP is known, it is mostly preferred in integer programming as an implementation of STP, since there exists a formulation of compact size, which turns out to be very effective in practice. It is known that the integrality gap of $\mathrm{…

Steiner树问题双向割公式的整数间隙下界

Lower bounds for the integrality gap of the bi-directed cut formulation of the Steiner Tree Problem

研究图上度量Steiner Tree问题，计算双向割（BCR）公式的整数间隙下界，引入Complete Metric（CM）模型以弥补BCR弱点，将Gap问题扩展至该领域，并通过设计启发式方法发现节点数<=10的实例实现已知最佳下界。

Steiner树的双向切割松弛具有小于2的整性间隙

The Bidirected Cut Relaxation for Steiner Tree has Integrality Gap Smaller than 2

针对Steiner树问题，本文证明双向切割松弛（BCR）的整性间隙上界为1.9988，首次严格表明其优于恰好为2的自然无向松弛。