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Transitive-Closure Spanners∗ Transitive-Closure Spanners∗ Arnab Bhattacharyya† Elena Grigorescu† Kyomin Jung† Sofya Raskhodnikova‡ David P. Woodruff§ Abstract Inapproximability. Our main technical contribution is We define the notion of a transitive-closure spanner of a di- a pair of strong inapproximability results. We resolve the ap- rected graph. Given a directed graph G = (V, E) and an in- proximability of 2-TC-spanners, showing that it is Θ(log n) unless P = NP . For constant k 3, we prove that the size teger k 1, a k-transitive-closure-spanner (k-TC-spanner) ≥ of is a≥ directed graph that has (1) the same of the sparsest k-TC-spanner is hard to approximate within G H = (V, EH ) 1−ǫ transitive-closure as G and (2) diameter at most k. These 2log n, for any ǫ > 0, unless NP DTIME(npolylog n). ⊆ spanners were studied implicitly in access control, property Our hardness result helps explain the difficulty in designing testing, and data structures, and properties of these spanners general efficient solutions for the applications above, and it have been rediscovered over the span of 20 years. We bring cannot be improved without resolving a long-standing open these areas under the unifying framework of TC-spanners. question in complexity theory. It uses an involved applica- We abstract the common task implicitly tackled in these di- tion of generalized butterfly and broom graphs, as well as verse applications as the problem of constructing sparse TC- noise-resilient transformations of hard problems, which may spanners. be of independent interest. We study the approximability of the size of the spars- Structural bounds. Finally, we study the size of est k-TC-spanner for a given digraph. Our technical contri- the sparsest TC-spanner for H-minor-free digraphs, which butions fall into three categories: algorithms for general di- include planar, bounded genus, and bounded tree-width graphs, inapproximability results, and structural bounds for graphs, explicitly investigated in applications above. We a specific graph family which imply an efficient algorithm show that every H-minor-free digraph has an efficiently con- with a good approximation ratio for that family. structible k-TC-spanner of size O˜(n). This implies an O˜(1)- Algorithms. We present two efficient deterministic al- approximation algorithm for this family. Furthermore, using gorithms that find k-TC-spanners of near optimal size. our insight that 2-TC-spanners yield property testers, we ob- 2 The first algorithm gives an O˜(n1−1/k)-approximation for tain a monotonicity tester with O(log n/ǫ) queries for any k > 2. Our method, based on a combination of convex poset whose transitive reduction is an H-minor free digraph. programming and sampling, yields the first sublinear ap- This improves and generalizes the previous Θ(√n log n/ǫ)- proximation ratios for (1) DIRECTED k-SPANNER, a well- query tester of Fischer et al (STOC, 2002). studied generalization of k-TC-SPANNER, and (2) its vari- ants CLIENT/SERVER DIRECTED k-SPANNER, and the k- 1 Introduction DIAMETER SPANNING SUBGRAPH. This resolves the main A spanner can be thought of as a sparse backboneof a graph open question of Elkin and Peleg (IPCO, 2001). The sec- that approximately preserves distances between every pair ond algorithm, specific to the k-TC-spanner problem, gives of vertices. More precisely, a subgraph H = (V, EH ) is an O˜(n/k2)-approximation. It shows that for k = Ω(√n), a k-spanner of G = (V, E) if for every pair of vertices our problem has a provably better approximation ratio than u, v V , the shortest path distance d (u, v) from u to ∈ H DIRECTED k-SPANNER and its variants. This algorithm also v in H is at most k dG(u, v). Since they were intro- resolves an open question of Hesse (SODA, 2003). duced by Peleg and Sch¨affer· [36] in the context of distributed computing, spanners for undirected graphs have been exten- ∗All omitted proofs and details appear in the full version [10]. sively studied. The tradeoff between the parameter k, called † Massachusetts Institute of Technology, USA. Email: {abhatt, the stretch, and the number of edges in a spanner is rela- elena g,kmjung}@mit.edu. A.B. was supported by National Science Foundation (NSF grants 0514771, 0732334, and 0728645) and DOE Com- tively well understood: for every k 1, any undirected graph on n vertices has a (2k 1)-spanner≥ with O(n1+1/k) putational Science Graduate Fellowship. E.G. was funded in part by NSF − grants CCR-0726525 and CCR-0829672. edges [6, 35, 47]. This is known to be tight for k = 1, 2, 3, 5 ‡Pennsylvania State University, USA. Email: [email protected]. and is conjectured to be tight for all k (see, for example a Supported by National Science Foundation (NSF grant CCF-0729171). survey by Zwick [50]). Undirected spanners have numerous §IBM Almaden Research Center, USA. Email: [email protected]. applications, such as efficient routing [15, 16, 38, 39, 46], 932 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. simulating synchronized protocols in unsynchronized net- et al. [12] study algorithms that might seem relevant to k- works [37], parallel and distributed algorithms for approx- TC-SPANNER. In [10], we explain why these algorithms do imating shortest paths [13, 14, 19], and algorithms for dis- not work for k-TC-SPANNER. tance oracles [9, 47]. For all constant k > 2 and ǫ (0, 1), it is impossi- ∈ In the directed setting, two notions of spanners have ble to approximate DIRECTED k-SPANNER within a factor 1−ǫ been considered in the literature: the direct generalization of of 2log n, assuming NP DTIME(npoly log n) [20]. More- the above definition [36] and roundtrip spanners [16, 39]. In over, [23] extend this result6⊆ to 3 k = O(n1−δ) for all this paper, we introduce a new definition of directed spanners δ (0, 1). Thus, according to Arora≤ and Lund’s classifica- ∈ that captures the notion that a spanner should have a small tion [31] of NP-hard problems, DIRECTED k-SPANNER is in diameter but preserve the connectivity of the original graph. class III, for 3 k = O(n1−δ). Moreover, [23] show that ≤ provingthat DIRECTED k-SPANNER is in class IV,that is, in- DEFINITION 1.1. (TC-SPANNER) Given a directed graph approximable within nδ for some δ (0, 1), would resolve a G = (V, E) and an integer k 1, a k-transitive-closure- ∈ ≥ long standing open question in complexity theory, and cause spanner (k-TC-spanner) is a directed graph H = (V, EH ) classes III and IV to collapse into a single class. with the following properties: (1) EH is a subset of theedges in the transitive closure of G. (2) For all vertices u, v V , 1.2 Our Contributions In this work we (1) bring several if d (u, v) < , then d (u, v) k. ∈ G ∞ H ≤ diverse applications, including property testing, access con- trol and data structures, under the unifying framework of Notice that a k-TC-spanner of G is just a directed k-spanner TC-spanners, (2) obtain bounds on the approximability of of the transitive-closure of G. Nevertheless, TC-spanners are interesting in their own right due to the numerous TC- k-TC-SPANNER,DIRECTED k-SPANNER and well-studied spanner-specific applications we present in Section 1.3. variants of these problems, and (3) construct sparse TC- One of the focuses of this paper is the study of the spanners for the family of H-minor free graphs, which in- computational problem of finding the size of the sparsest clude planar, bounded-treewidth, and bounded genus graphs. Table 1 summarizes our results on the approximability of - k-TC-spanner for a given digraph, referred to as k-TC- k TC-SPANNER. SPANNER. It is a special case of the problem of finding the Algorithms for k-TC-SPANNER and related prob- size of the sparsest directed spanner, called DIRECTED k- lems. We present two deterministic polynomial time ap- SPANNER, that has been previously studied. Both problems are NP-hard (proofs appear in the full version [10]). proximation algorithms for k-TC-SPANNER. Our first al- gorithm uses a new combination of convex programming 1−1/k 1.1 Related Work Thorup [42] considered a special case and sampling, and gives an O((n log n) )-ratio for k- TC-SPANNER. Moreover, our method yields the same ap- of TC-spanners of graphs G that have at most twice as proximation ratio for DIRECTED -SPANNER and its well- many edges as G, and conjectured that for all directed k studied variants: CLIENT/SERVER DIRECTED -SPANNER, graphs G on n nodes there are such TC-spanners with stretch k and -DIAMETER SPANNING SUBGRAPH (see [21] for def- polylogarithmic in n. He proved his conjecture for planar k graphs [43], but later Hesse [30] gave a counterexample to initions). This resolves the open question of finding a sub- Thorup’s conjecture for general graphs. TC-spanners were linear approximation ratio for these problems for k > 3, de- also studied for directed trees: implicitly in [5, 8, 11, 17, 49] scribed as a “challenging direction” for research on directed and explicitly in [44]. For the directed line, [5] (and later, [8]) spanners by Elkin and Peleg [22]. Our algorithm for k = 3 is arguably simpler than the 2/3 -approximation showed that the size of the sparsest k-TC-spanner is Θ(n O(n polylog n) algorithm of [22]. λ (n)), where λ (n) is the kth-row inverse Ackermann· k k ˜ 2 function.
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