Transitive-Closure Spanners

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, E H ) 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 size approximating the optimum. The first algorithm gives anÕ(n 1−1/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 anÕ(n/k 2 )-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 = N P . For constant k ≥ 3, we prove that the size of the sparsest k-TC-spanner is hard to approximate within 2 log 1− n , for any > 0, unless NP ⊆ DTIME(n polylog 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 longstanding 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 constructable k-TC-spanner of sizeÕ(n), which implies anÕ(1)-approximation algorithm for this family. Furthermore, using our insight that 2-TC-spanners yield property testers, we obtain a monotonicity tester with O(log 2 n...

Testing for forbidden order patterns in an array

“…The study of combinatorial properties was initiated by Goldreich, Goldwasser, and Ron [27]. Works on string properties related to forbidden or occurring patterns in labeled posets include the papers on testing sortedness [9,21] and the lower bound of Fischer [22], and many others, see for example, [3,7,24,37], and citations therein. The problem of testing hereditary properties of permutations has been studied before by Hoppen and coworkers [31] under the rectangular distance (discrepancy of intervals) and by Klimošová and Král [36] under Kendall's tau-distance (the normalized number of transpositions).…”

“…The first testing algorithm for monotonicity was developed by Ergün and coworkers and a matching lower bound was given by Fischer . Later Bhattacharyya and coworkers developed a very simple and elegant tester for this property.…”

“…The study of combinatorial properties was initiated by Goldreich, Goldwasser, and Ron . Works on string properties related to forbidden or occurring patterns in labeled posets include the papers on testing sortedness and the lower bound of Fischer , and many others, see for example, , and citations therein.…”

“…Another line of research relevant to our problem (mostly to the extension to partial orders discussed in the previous subsection) is monotonicity testing . Recently, Chakrabarty and Seshadhri gave an optimal tester for this problem on the hypercube and on hypergrids .…”

Approximation Algorithms and an Integer Program for Multi-level Graph Spanners

“…They provide aÕ(n 2 3 )-approximation algorithm for the directed kspanner problem for k ≥ 1, which is the first sublinear approximation algorithm for arbitrary edge lengths. Bhattacharyya et al [6] provide a slightly different formulation to approximate t-spanners as well as other variations of this problem. They provide a polynomial time O((n log n) 1− 1 k )-approximation algorithm for the directed k-spanner problem.…”