[论文解读] Transitive-closure spanners

Arnab Bhattacharyya, Elena Grigorescu+3 more|arXiv (Cornell University)|Jan 4, 2009

Complexity and Algorithms in Graphs参考文献 52被引用 50

一句话总结

本文引入了传递闭包的简并图（TC-spanners）作为访问控制、属性测试和数据结构等问题的统一框架。该文提出了高效构造稀疏 k-TC-spanner 的算法，首次实现了相关问题的次线性近似比，证明了强不可近似性界，并为 H-无小图有向图建立了结构上的 O(n) 大小界，从而实现了更优的属性测试。

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 O(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 O(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-en, for any e > 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 con-structible k-TC-spanner of size O(n). This implies an O(1)-approximation algorithm for this family. Furthermore, using our insight that 2-TC-spanners yield property testers, we obtain a monotonicity tester with O(log2n/e) queries for any poset whose transitive reduction is an H-minor free digraph. This improves and generalizes the previous θ(√n log n/e)-query tester of Fischer et al (STOC, 2002).

研究动机与目标

• 将访问控制、属性测试和数据结构等多样化应用统一于传递闭包简并图的框架之下。
• 研究为一般有向图构造最稀疏 k-TC-spanner 的近似可解性。
• 解决关于 k-TC-spanner 及其相关问题（如有向 k-简并图和 k-直径生成子图）的近似比的开放问题。
• 为 2-TC-spanners 和常数 k ≥ 3 建立强不可近似性结果。
• 推导 H-无小图有向图的 TC-spanner 大小的结构界，从而实现高效的近似与更优的属性测试。

提出的方法

• 使用凸规划与随机采样，设计出针对 k > 2 的确定性 O(n^{1-1/k})-近似算法。
• 应用一种针对 k-TC-spanners 的专用算法，实现 O(n/k²)-近似，优于 k = Ω(√n) 时的通用 k-简并图方法。
• 通过新颖构造使用广义的蝴蝶图与扫帚图，证明强不可近似性下界。
• 利用对困难问题的抗噪声变换，证明除非 NP ⊆ DTIME(n^{polylog n})，否则 k ≥ 3 时近似 k-TC-spanners 的难度不低于 2^{log^{1-ε} n}。
• 利用结构洞察，为 H-无小图有向图构造出大小为 O(n) 的高效 k-TC-spanners。
• 利用 2-TC-spanners 构造出针对 H-无小图传递约化的偏序集的单调性测试器，查询次数为 O(log²n / ε)。

实验结果

研究问题

• RQ1 为一般有向图构造最稀疏 k-TC-spanner 的近似可解性如何？能否实现次线性近似比？
• RQ2 Elkin 和 Peleg 关于有向 k-简并图的开放问题能否通过次线性近似方法解决？
• RQ3 2-TC-spanners 以及常数 k ≥ 3 的最强不可近似性界是什么？
• RQ4 H-无小图有向图是否具有稀疏且可高效构造的 k-TC-spanner，且近似比为常数？
• RQ5 TC-spanner 构造能否带来更优的属性测试，特别是针对单调性测试？

主要发现

• 本文首次为 k > 2 实现了 O(n^{1-1/k}) 的次线性近似比，解决了 Elkin 和 Peleg 提出的开放问题。
• 当 k = Ω(√n) 时，k-TC-spanner 的 O(n/k²)-近似算法在近似比上优于现有有向 k-简并图及其变体的方法。
• 本文证明 2-TC-spanners 的近似比为 θ(log n)，除非 P = NP，从而建立了紧致的不可近似性界。
• 对于常数 k ≥ 3，最稀疏 k-TC-spanner 的大小在任何 ε > 0 下，难以在 2^{log^{1-ε} n} 范围内近似，除非 NP ⊆ DTIME(n^{polylog n})。
• 每个 H-无小图有向图都存在一个可高效构造的 k-TC-spanner，其大小为 O(n)，从而为该类图族提供了 O(1)-近似算法。
• 该框架使得针对其传递约化为 H-无小图的偏序集的单调性测试器，查询次数为 O(log²n / ε)，优于先前的 θ(√n log n / ε) 查询界。