Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs si, ti, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi gave a primal-dual constant-factor approximation algorithm for this problem. Until this work, the only constant-factor approximations we know are via linear programming relaxations. In this paper, we consider the following greedy algorithm: Given terminal pairs in a metric space, a terminal is active if its distance to its partner is non-zero. Pick the two closest active terminals (say si, tj), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat.} It has long been open to analyze this greedy algorithm. Our main result shows that this algorithm is a constant-factor approximation. We use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest.