Refined Vertex Sparsifiers of Planar Graphs

We study the following version of cut sparsification. Given a large edge-weighted network G with k terminal vertices, compress it into a smaller network H with the same terminals, such that every minimum terminal cut in H approximates the corresponding one in G, up to a factor q >= 1 that is called the quality. (The case q = 1 is known also as a mimicking network.) We provide new insights about the structure of minimum terminal cuts, leading to new results for cut sparsifiers of planar graphs. Our first contribution identifies a subset of the minimum terminal cuts, which we call elementary, that generates all the others. Consequently, H is a cut sparsifier if and only if it preserves all the elementary terminal cuts (up to this factor q). Our second and main contribution is to refine the known bounds in terms of gamma = gamma(G), which is defined as the minimum number of faces that are incident to all the terminals in a planar graph G. We prove that the number of elementary terminal cuts is O((2k/gamma)(2 gamma)) (compared to O(2(k)) terminal cuts) and furthermore obtain a mimicking network of size O(gamma 2(2 gamma) k(4)), which is near-optimal as a function of gamma. Our third contribution is a duality between cut sparsification and distance sparsification for certain planar graphs, when the sparsifier H is required to be a minor of G. This duality connects problems that were previously studied separately, implying new results, new proofs of known results, and equivalences between open gaps.