基于单级求解器的多级图形草图
Multi-Level Graph Sketches via Single-Level Solvers
摘要：
给定一个无向加权图$G(V,E)$，终端集$T\subset V$上的受约束草图是一个子图$G'$，它连接终端顶点，同时满足给定的一组约束。例子包括Steiner树（保持$T$之间的连通性）和subsetwise spectrum（保持最短路径距离达到拉伸因子）。多级约束端子草图是一种概括，其中端子顶点需要不同的服务级别或等级，并且每个端子对至少与两个端子的最低所需级别的边连接。本文给出了一个灵活的程序计算一个广泛的类的多级图草图，其中包括多级图spectrometer，斯坦纳树，$k$ -连接的子图作为几个特殊情况。所提出的程序是模块化的，即，它依赖于单级求解器模块的可用性（无论是Oracle还是近似算法）。一个关键的结果是，$\ell$ -级约束的终端草图可以计算与$\log\ell$查询的求解器模块，同时产生可行的解决方案与近似保证独立于$\ell$。此外，提出了一种新的多项式时间算法计算的subsetwise。我们发现，$k\in\N$，$\eps>0$和$T\subset V$，有一个subsetwise $(2k-1)(1+\eps)$ -与总重量$O(|T|^\frac1kW(\ST(G,T)))$，其中$W(\ST(G,T))$是在子集$T$的斯坦纳树的$G$的重量。这是第一个算法和重量保证的乘法子集的非平面图。数值实验也证明了所提出算法的性能。
英文摘要：
Given an undirected weighted graph $G(V,E)$, a constrained sketch over a terminal set $T\subset V$ is a subgraph $G'$ that connects the terminal vertices while satisfying a given set of constraints. Examples include Steiner trees (preserving connectivity among $T$) and subsetwise spanners (preserving shortest path distances up to a stretch factor). Multi-level constrained terminal sketches are generalizations in which terminal vertices require different levels or grades of service and each terminal pair is connected with edges of at least the minimum required level of the two terminals. This paper gives a flexible procedure for computing a broad class of multi-level graph sketches, which encompasses multi-level graph spanners, Steiner trees, and $k$--connected subgraphs as a few special cases. The proposed procedure is modular, i.e., it relies on availability of a single-level solver module (be it an oracle or approximation algorithm). One key result is that an $\ell$--level constrained terminal sketch can be computed with $\log\ell$ queries of the solver module while producing feasible solutions with approximation guarantees independent of $\ell$. Additionally, a new polynomial time algorithm for computing a subsetwise spanner is proposed. We show that for $k\in\N$, $\eps>0$, and $T\subset V$, there is a subsetwise $(2k-1)(1+\eps)$--spanner with total weight $O(|T|^\frac1kW(\ST(G,T)))$, where $W(\ST(G,T))$ is the weight of the Steiner tree of $G$ over the subset $T$. This is the first algorithm and weight guarantee for a multiplicative subsetwise spanner for nonplanar graphs. Numerical experiments are also done to illustrate the performance of the proposed algorithms.
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