[论文解读] Approximating Directed Steiner Problems via Tree Embedding

Rothvoß, Thomas|arXiv (Cornell University)|Nov 23, 2011

Complexity and Algorithms in Graphs参考文献 39被引用 33

一句话总结

该论文提出了一种针对有向斯坦纳树（DST）问题的多项式时间 |X|ε-近似算法，通过使用 O(ℓ)-轮 Lasserre 层次强化基于流的线性规划（LP）松弛，实现了在 ℓ 层有向无环图中的 O(ℓ log |X|) 整体间隙。该方法结合一种新颖的基于采样的舍入方案与路径覆盖的概率分析，在 n^{O(log|X|)} 时间内实现了 O(log³|X|) 的近似比，与目前已知性能最佳的贪心算法相当。

ABSTRACT

Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H subseteq G that connects r to each terminal t in T by k edge-disjoint r, t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k=1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14], by Laekhanukit [SODA'14] and in a survey by Kortsarz and Nutov [Handbook of Approximation Algorithms and Metaheuristics], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in a special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k=2 and h=2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D*k^{D-1}*log(n))-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r, t-paths, each of length at most D, for every terminal t in T. For the case k=1 (DST), our algorithm yields an approximation ratio of O(D*log(h)), thus implying an O(log^3(h))-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Our algorithm is based on an LP-formulation that allows us to embed a solution to a tree-instance of GST, which does not preserve connectivity. We show, however, that one can randomly extract a solution of k-DST from the tree-instance of GST. Our algorithm is almost tight when k and D are constants since the case that k=1 and D=3 is NP-hard to approximate to within a factor of O(log(h)), and our algorithm archives the same approximation ratio for this special case. We also remark that the k^{1/4-epsilon}-hardness instance of k-DST is a DAG with 6 layers, and our algorithm gives O(k^5*log(n))-approximation for this special case. Consequently, as our algorithm works for general graphs, we obtain an O(D*k^{D-1}*log(n))-approximation algorithm for a D-shallow instance of the k edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k edgedisjoint paths.

研究动机与目标

• 为解决有向斯坦纳树（DST）问题自然 LP 松弛的高整体间隙问题，该问题在 5 层时即达到 Ω(√|X|) 的下界。
• 探索更强的 LP/SDP 层次（如 Lasserre 层次）是否能显著降低该间隙并获得更好的近似保证。
• 提供一种多项式时间的 DST 近似算法，其性能与或优于目前已知的贪心算法，例如 Charikar 等人提出的 O(log³|X|) 近似。
• 证明 O(ℓ)-轮 Lasserre 强化在 ℓ 层有向无环图中可实现 O(ℓ log |X|) 的整体间隙，这是 DST 的一个关键结构设定。
• 建立 Lasserre 松弛与基于概率路径采样的有效舍入之间的联系，利用解的凸组合表示形式。

提出的方法

• 使用基于路径 P 的变量 yP 和边容量变量 y{e} 的基于流的 LP 松弛来表示 DST 问题。
• 对 LP 应用 O(ℓ)-轮 Lasserre 层次，生成一个解 Y，该解是大小不超过 ℓ 的集合上的整数解的凸组合。
• 利用 Lasserre 分解定理与 LP 约束，证明路径变量 yP 满足 yP = 1 当且仅当 P 是一条有效的 r-s 路径。
• 设计一种随机舍入方案，独立地以与 yP 成比例的概率采样路径 T，确保终端连通性的期望值 E[Z] = 1。
• 将每个节点的采样路径期望数量控制在 O(n) 以内，每条采样路径的 Lasserre 查询次数控制在 O(n²) 以内，从而保证多项式时间的 oracle 访问。
• 利用 Janson 类型不等式与条件期望界，证明 Pr[Z ≥ 1] ≥ 1/(ℓ + 1)，其中 Z 是覆盖终端 s 的路径数量，从而保证常数连通性概率。

实验结果

研究问题

• RQ1 O(ℓ)-轮 Lasserre 层次是否能降低 ℓ 层有向无环图中 DST 问题自然 LP 松弛的整体间隙？
• RQ2 Lasserre 层次是否能实现一种多项式时间的 DST 近似算法，其近似比与目前已知性能最佳的贪心算法相当？
• RQ3 在基于 Lasserre 的路径采样方案下，终端的期望连通概率是多少？是否可被下界控制为常数？
• RQ4 鉴于依赖于分解定理，Lasserre 层次是否严格强于较弱的层次（如 Sherali-Adams 或 Lovász-Schrijver）用于 DST 问题？
• RQ5 尽管变量数量呈指数级增长，是否仍可使用能返回单个路径变量 yP 的多项式时间 oracle 来高效实现该算法？

主要发现

• O(ℓ)-轮 Lasserre 强化对基于流的 DST LP 松弛，在 ℓ 层有向无环图中实现了 O(ℓ log |X|) 的整体间隙。
• 对于任意常数 ε > 0，通过设置 ℓ = log|X|，该方法可实现多项式时间的 |X|ε-近似算法。
• 该算法在时间 n^{O(log|X|)} 内实现了 O(log³|X|) 的近似比，与 Charikar 等人提出的贪心算法在时间-近似比权衡上达到最优。
• 随机舍入方案确保每个终端的连通概率至少为 1/(ℓ + 1)，且解的期望代价为 LP 值的 O(ℓ log |X|) 倍。
• 假设可访问单个路径变量 yP 的 oracle，Lasserre 查询次数与运行时间的期望值均为 n 的多项式函数。
• 该结果关键依赖于 Lasserre 分解定理，而该定理在 Sherali-Adams 等较弱层次中不成立，暗示了性能上可能存在分离。