## Abstract

We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TC-spanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We study the approximability of the size of the sparsest k-TC-spanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs, inapproximability results, and structural bounds for a specific graph family which imply an efficient algorithm with a good approximation ratio for that family. Algorithms. We present two efficient deterministic algorithms that find k-TC-spanners of near optimal size. The first algorithm gives an Õ(n1-1/k)-approximation for k > 2. Our method, based on a combination of convex programming and sampling, yields the first sublinear approximation ratios for (1) DIRECTED k-SPANNER, a well-studied generalization of k-TC-SPANNER, and (2) its variants CLIENT/SERVER DIRECTED k-SPANNER, and the k-DIAMETER SPANNING SUBGRAPH. This resolves the main open question of Elkin and Peleg (IPCO, 2001). The second algorithm, specific to the k-TC-spanner problem, gives an Õ(n/k2)-approximation. It shows that for k = Ω(√n), our problem has a provably better approximation ratio than DIRECTED k-SPANNER and its variants. This algorithm also resolves an open question of Hesse (SODA, 2003). Inapproximability. Our main technical contribution is a pair of strong inapproximability results. We resolve the approximability of 2-TC-spanners, showing that it is Θ(log n) unless P = NP. For constant k ≥ 3, we prove that the size of the sparsest k-TC-spanner is hard to approximate within 2log1-∈ n, for any ∈ > 0, unless NP ⊆ DTIME(npolylog n). Our hardness result helps explain the difficulty in designing general efficient solutions for the applications above, and it cannot be improved without resolving a long-standing open question in complexity theory. It uses an involved application of generalized butterfly and broom graphs, as well as noise-resilient transformations of hard problems, which may be of independent interest. Structural bounds. Finally, we study the size of the sparsest TC-spanner for H-minor-free digraphs, which include planar, bounded genus, and bounded tree-width graphs, explicitly investigated in applications above. We show that every H-minor-free digraph has an efficiently constructible k-TC-spanner of size Õ(n). This implies an Õ(1)-approximation algorithm for this family. Furthermore, using our insight that 2-TC-spanners yield property testers, we obtain a monotonicity tester with O(log2 n/∈) queries for any poset whose transitive reduction is an H-minor free digraph. This improves and generalizes the previous Θ(√n log n/∈)-query tester of Fischer et al (STOC, 2002). Copyright © by SIAM.