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

TL;DR

This paper introduces an efficient algorithm for solving a specific LP relaxation of the Steiner tree problem in quasi-bipartite graphs, leading to improved approximation ratios.

Contribution

It provides the first efficient exact algorithm for the directed component relaxation in quasi-bipartite graphs, enhancing Steiner tree approximation methods.

Findings

Efficient exact algorithm for DCR in quasi-bipartite graphs.

Achieves a 73/60-approximation ratio for the Steiner tree problem.

Proposes a simple sampling algorithm with a performance guarantee slightly better than 77/60.

Abstract

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 efficient 73/60-approximation for quasi-bipartite Steiner tree instances. We also present a particularly simple (BCR)-based random sampling algorithm that achieves a performance guarantee slightly better than 77/60.