We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph $G$ and a root node $r$, we seek a single spanning tree $T$ of minimum {\em stretch}, where the stretch of $T$ is defined to be the maximum ratio, over all terminal sets $X$, of the cost of the minimal sub-tree $T_X$ of $T$ that connects $X$ to $r$ to the cost of an optimal Steiner tree connecting $X$ to $r$ in $G$. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications.
We provide a polynomial time \ust\ construction for general graphs with $2^{O(\sqrt{\log n})}$-stretch. We also give a polynomial time $\polylog(n)$-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first subpolynomial-stretch ($o(n^ε)$ for any $ε> 0$) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs.
Our main result is that the Steiner Point Removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa (2006). Specifically, we prove that for every edge-weighted graph $G = (V,E,w)$ and a subset of terminals $T \subseteq V$, there is a graph $G'=(T,E',w')$ that is isomorphic to a minor of $G$, such that for every two terminals $u,v\in T$, the shortest-path distances between them in $G$ and in $G'$ satisfy $d_{G,w}(u,v) \le d_{G',w'}(u,v) \le O(\log^5|T|) \cdot d_{G,w}(u,v)$. Our existence proof …
The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or, Steiner nodes), where $|T|=k$, find a minimum-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized PTAS for the weighted problem. We obtain similar results for the $k$-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a…
This paper summarizes the work on implementing few solutions for the Steiner Tree problem which we undertook in the PAAL project. The main focus of the project is the development of generic implementations of approximation algorithms together with universal solution frameworks. In particular, we have implemented Zelikovsky 11/6-approximation using local search framework, and 1.39-approximation by Byrka et al. using iterative rounding framework. These two algorithms are experimentally compared with greedy 2-approximation, with exact but exponential time Dreyfus-Wagner algorithm, as well as with…
Given a complete graph $G=(V,E)$, with nonnegative edge costs, two subsets $R \subset V$ and $R^{\prime} \subset R$, a partition $\mathcal{R}=\{R_1,R_2,\ldots,R_k\}$ of $R$, $R_i \cap R_j=ϕ$, $i \neq j$ and $\mathcal{R}^{\prime}=\{R^{\prime}_1,R^{\prime}_2,\ldots,R^{\prime}_k\}$ of $R^{\prime}$, $R^{\prime}_i \subset R_i$, a clustered Steiner tree is a tree $T$ of $G$ that spans all vertices in $R$ such that $T$ can be cut into $k$ subtrees $T_i$ by removing $k-1$ edges and each subtree $T_i$ spanning all vertices in $R_i$, $1 \leq i \leq k$. The cost of a clustered Steiner tree is defined to …
University of Trieste（的里雅斯特大学）；CWI（荷兰国家数学与计算机科学研究中心）；University of Bremen（不来梅大学）；La Sapienza University of Rome（罗马第一大学）；INRIA-Erable（法国国家信息与自动化研究所Erable团队）；Vrije Universiteit（阿姆斯特丹自由大学）
We introduce a novel measure for quantifying the error in input predictions. The error is based on a minimum-cost hyperedge cover in a suitably defined hypergraph and provides a general template which we apply to online graph problems. The measure captures errors due to absent predicted requests as well as unpredicted actual requests; hence, predicted and actual inputs can be of arbitrary size. We achieve refined performance guarantees for previously studied network design problems in the online-list model, such as Steiner tree and facility location. Further, we initiate the study of learning-…
University of New South Wales（新南威尔士大学）；Fudan University（复旦大学）
The Binary Space Partitioning-Tree~(BSP-Tree) process was recently proposed as an efficient strategy for space partitioning tasks. Because it uses more than one dimension to partition the space, the BSP-Tree Process is more efficient and flexible than conventional axis-aligned cutting strategies. However, due to its batch learning setting, it is not well suited to large-scale classification and regression problems. In this paper, we develop an online BSP-Forest framework to address this limitation. With the arrival of new data, the resulting online algorithm can simultaneously expand the space…
