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

Abstract

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 undirected 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 ⊆ 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 ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal Steiner tree T for S and that has size polynomial in g and | E ( T )|. — 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 ⊆ V ( G ), one can in polynomial time find a set F ⊆ 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 that has size polynomial in | C |. In the language of parameterized complexity, these results imply the first polynomial kernels for S teiner T ree and S teiner F orest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (E dge ) M ultiway C ut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an undirected plane graph G with positive edge weights, a designated face f bounded by a simple cycle of weight w ( f ), and an accuracy parameter ε > 0, uncovers a set F ⊆ E ( G ) of total weight at most poly(ε -1 ) w ( f ) that, for any set of terminal pairs that lie on f , contains a Steiner forest within additive error ε w ( f ) from the optimal Steiner forest.