Optimal Vertex-Cut Sparsification of Quasi-Bipartite Graphs

Abstract

In vertex-cut sparsification, given a graph G = (V, E) with a terminal set T ⊆ V , we wish to construct a graph G ′ = (V ′ , E ′ ) with T ⊆ V ′ , such that for every two sets of terminals A, B ⊆ T , the size of a minimum (A, B)-vertex-cut in G ′ is the same as in G. In the most basic setting, G is unweighted and undirected, and we wish to bound the size of G ′ by a function of k = |T |. Kratsch and Wahlström [JACM 2020] proved that every graph G (possibly directed), admits a vertex-cut sparsifier G ′ with O(k 3 ) vertices, which can in fact be constructed in randomized polynomial time.We study (possibly directed) graphs G that are quasi-bipartite, i.e., every edge has at least one endpoint in T , and prove that they admit a vertex-cut sparsifier with O(k 2 ) edges and vertices, which can in fact be constructed in deterministic polynomial time. In fact, this bound naturally extends to all graphs with a small separator into bounded-size sets. Finally, we prove information-theoretically a nearly-matching lower bound, i.e., that Ω(k 2 ) edges are required to sparsify quasi-bipartite undirected graphs.