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Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also 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.