A PTAS for the Steiner Forest Problem in Doubling Metrics

We achieve a (randomized) polynomial-time approximation scheme (PTAS) for the Steiner Forest Problem in doubling metrics. Before our work, a PTAS is given only for the Euclidean plane in [FOCS 2008: Borradaile, Klein and Mathieu]. Our PTAS also shares similarities with the dynamic programming for sparse instances used in [STOC 2012: Bartal, Gottlieb and Krauthgamer] and [SODA 2016: Chan and Jiang]. However, extending previous approaches requires overcoming several non-trivial hurdles, and we make the following technical contributions. (1) We prove a technical lemma showing that Steiner points have to be "near" the terminals in an optimal Steiner tree. This enables us to define a heuristic to estimate the local behavior of the optimal solution, even though the Steiner points are unknown in advance. This lemma also generalizes previous results in the Euclidean plane, and may be of independent interest for related problems involving Steiner points. (2) We develop a novel algorithmic technique known as "adaptive cells" to overcome the difficulty of keeping track of multiple components in a solution. Our idea is based on but significantly different from the previously proposed "uniform cells" in the FOCS 2008 paper, whose techniques cannot be readily applied to doubling metrics.