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 = (V H , E H ) such that (1) V ⊆ V H 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 S k , of its sparsest Steiner k-TC-spanner.
We present a nearly tight lower bound on S 2 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 S k for d-dimensional posets.
KeywordsDirected Graph Linear Programming Relaxation Full Version Dual Program Dual Linear Program
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- 2.Atallah, M.J., Blanton, M., Fazio, N., Frikken, K.B.: Dynamic and efficient key management for access hierarchies. ACM Trans. Inf. Syst. Secur. 12(3) (2009)Google Scholar
- 3.Awerbuch, B.: Communication-time trade-offs in network synchronization. In: PODC, pp. 272–276 (1985)Google Scholar
- 4.Bhattacharyya, A., Grigorescu, E., Jha, M., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Lower bounds for local monotonicity reconstruction from transitive-closure spanners. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 448–461. Springer, Heidelberg (2010)CrossRefGoogle Scholar
- 5.Bhattacharyya, A., Grigorescu, E., Jung, K., Raskhodnikova, S., Woodruff, D.P.: Transitive-closure spanners. In: Mathieu, C. (ed.) SODA, pp. 932–941. SIAM, Philadelphia (2009)Google Scholar
- 11.Jha, M., Raskhodnikova, S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. Electronic Colloquium on Computational Complexity (ECCC) TR11-057 (2011)Google Scholar