Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k, uncovers a set F subset of E(G) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f. We apply this general theorem to prove that: given an unweighted graph G embedded on a surface of genus g and a terminal set S subset of V (G), one can in polynomial time find a set F subset of E(G) that contains an optimal Steiner tree T for S and that has size polynomial in g and vertical bar E(T)vertical bar; an analogous result holds for an optimal Steiner forest for a set S of terminal pairs; given an unweighted planar graph G and a terminal set S subset of V (G), one can in polynomial time find a set F subset of E(G) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different terminals from S) and has size polynomial in vertical bar C vertical bar. In the language of parameterized complexity, these results imply the first polynomial kernels for STEINER TREE and STEINER FOREST on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (EDGE) MULTIWAY CUT on planar graphs (parameterized by the size of the cutset). STEINER TREE and similar "subset" problems were identified in [1] as important to the quest to widen the reach of the theory of bidimensionality [2, 3]. Therefore, our results can be seen as a leap forward to achieve this broader goal. Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an edge-weighted planar graph G, a designated face f bounded by a simple cycle of weight w(f), and an accuracy parameter epsilon > 0, uncovers a set F subset of E(G) of total weight at most poly(epsilon(-1)) w(f) that, for any set of terminal pairs that lie on f, contains a Steiner forest within additive error epsilon w(f) from the optimal Steiner forest. This result deepens the understanding of the recent framework of approximation schemes for network design problems on planar graphs ([4, 5], and later works) by explaining the structure of the solution space within a brick of the so-called mortar graph - the central notion of this framework.