Steiner Transitive-Closure Spanners of Low-Dimensional Posets

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k-transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (VH , EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V , the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we concentrate on Steiner TC-spanners of directed acyclic graphs or, equivalently, partially ordered sets. We study the relationship between the dimension of a poset and the size, denoted Sk, of its sparsest Steiner k-TC-spanner. We present a nearly tight lower bound on S2 for d-dimensional directed hypergrids. Our bound is derived from an explicit dual solution to a linear programming relaxation of the 2-TC-spanner problem. We also give an efficient construction of Steiner 2-TC-spanners, of size matching the lower bound, for all low-dimensional posets. Finally, we present a nearly tight lower bound on Sk for d-dimensional posets.