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 property that the elements of each pair belong to the same connected component of the forest. The current best approximation factor for Steiner Forest is 2, which is achieved by the classical primal-dual algorithm; improving on this factor is a big open problem in the area. Motivated by this open problem, we study an LP relaxation for Steiner Forest that generalizes the well-studied Bidirected Cut Relaxation for Steiner Tree. We prove that this relaxation has several promising properties. Among them, it is possible to round any half-integral LP solution to a Steiner Forest instance while increasing the cost by at most a factor 16/9. To prove this result we introduce a novel recursive densest-subgraph contraction algorithm.

翻译：Steiner森林问题是Steiner树问题的重要推广。给定一个具有非负边成本的无向图及若干顶点对集合，任务是计算一个成本最低的森林，使得每个顶点对中的两个顶点均位于该森林的同一连通分量内。当前Steiner森林问题的最佳近似比为2，这一结果由经典的原对偶算法实现；改进该近似比是该领域的重要开放问题。受此开放问题启发，我们研究了一种Steiner森林的线性规划松弛方法，该方法推广了已被深入研究的Steiner树双向割松弛。我们证明该松弛具有若干良好性质：其中任意半整数线性规划解均可通过舍入操作转化为Steiner森林实例，且成本增加不超过16/9倍。为证明该结果，我们提出了一种新颖的递归稠密子图收缩算法。