Split and join: strong partitions and Universal Steiner trees for graphs

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 stretch, where the stretch of T is defined to be the maximum ratio, over all subsets of terminals X, of the ratio of the cost of the sub-tree TX that connects r to X to the cost of an optimal Steiner tree connecting X to r. 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 2O(√log n)-stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block in our algorithm is a hierarchy of graph partitions, each of which guarantees small strong cluster diameter and bounded local neighbourhood intersections. Our partition hierarchy for minor-free graphs is based on the solution to a cluster aggregation problem that may be of independent interest. To our knowledge, this is the first sub-linear UST result for general graphs, and the first polylogarithmic construction for minor-free graphs

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$

Computational metric embeddings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 141-145).We study the problem of computing a low-distortion embedding between two metric spaces. More precisely given an input metric space M we are interested in computing in polynomial time an embedding into a host space M' with minimum multiplicative distortion. This problem arises naturally in many applications, including geometric optimization, visualization, multi-dimensional scaling, network spanners, and the computation of phylogenetic trees. We focus on the case where the host space is either a euclidean space of constant dimension such as the line and the plane, or a graph metric of simple topological structure such as a tree. For Euclidean spaces, we present the following upper bounds. We give an approximation algorithm that, given a metric space that embeds into R1 with distortion c, computes an embedding with distortion c(1) [delta]3/4 (A denotes the ratio of the maximum over the minimum distance). For higher-dimensional spaces, we obtain an algorithm which, for any fixed d > 2, given an ultrametric that embeds into Rd with distortion c, computes an embedding with distortion co(1). We also present an algorithm achieving distortion c logo(1) [delta] for the same problem. We complement the above upper bounds by proving hardness of computing optimal, or near-optimal embeddings. When the input space is an ultrametric, we show that it is NP-hard to compute an optimal embedding into R2 under the ... norm. Moreover, we prove that for any fixed d > 2, it is NP-hard to approximate the minimum distortion embedding of an n-point metric space into Rd within a factor of Q(n1/(17d)). Finally, we consider the problem of embedding into tree metrics. We give a 0(1)approximation algorithm for the case where the input is the shortest-path metric of an unweighted graph.(cont.) For general metric spaces, we present an algorithm which, given an n-point metric that embeds into a tree with distortion c, computes an embedding with distortion (clog n)o ... . By composing this algorithm with an algorithm for embedding trees into R1, we obtain an improved algorithm for embedding general metric spaces into R1.by Anastasios Sidiropoulos.Ph.D

Efficient geometric algorithms for preference top-k queries, stochastic line arrangements, and proximity problems

University of Minnesota Ph.D. dissertation. June 2017. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); x, 150 pages.Problems arising in diverse real-world applications can often be modeled by geometric objects such as points, lines, and polygons. The goal of this dissertation research is to design efficient algorithms for such geometric problems and provide guarantees on their performance via rigorous theoretical analysis. Three related problems are discussed in this thesis. The first problem revisits the well-known problem of answering preference top-k queries, which arise in a wide range of applications in databases and computational geometry. Given a set of n points, each with d real-valued attributes, the goal is to organize the points into a suitable data structure so that user preference queries can be answered efficiently. A query consists of a d-dimensional vector w, representing a user's preference for each attribute, and an integer k, representing the number of data points to be retrieved. The answer to a query is the k highest-scoring points relative to w, where the score of a point, p, is designed to reflect how well it captures, in aggregate, the user's preferences for the different attributes. This thesis contributes efficient exact solutions in low dimensions (2D and 3D), and a new sampling-based approximation algorithm in higher dimensions. The second problem extends the fundamental geometric concept of a line arrangement to stochastic data. A line arrangement in the plane is a partition of the plane into vertices, edges, and faces. Surprisingly, diverse problems, including the preference top-k query and k-order Voronoi Diagram, essentially boil down to answering questions about the set of k-topmost lines at some abscissa. This thesis considers line arrangements in a new setting, where each line has an associated existence probability representing uncertainty that is inherent in real-world data. An upper-bound is derived on the expected number of changes in the set of k-topmost lines, taken over the entire x-axis, and a worst-case upper bound is given for k = 1. Also, given is an efficient algorithm to compute the most likely k-topmost lines in the arrangement. Applications of this problem including the most likely Voronoi Diagram in R^1 and stochastic preference top-k query are discussed. The third problem discussed is geometric proximity search in both the stochastic setting and the query-retrieval setting. Under the stochastic setting, the thesis considers two fundamental problems, namely, the stochastic closest pair problem and the k most likely nearest neighbor search. In both problems, the data points are assumed to lie on a tree embedded in R^2 and distances are measured along the tree (a so-called tree space). For the former, efficient solutions are given to compute the probability that the closest pair distance of a realization of the input is at least l and to compute the expected closest pair distance. For the latter, the thesis generalizes the concept of most likely Voronoi Diagram from R^1 to tree space and bounds its combinatorial complexity. A data structure for the diagram and an algorithm to construct it are also given. For the query-retrieval version which is considered in R^2, the goal is to retrieve the closest pair within a user-specified query range. The contributions here include efficient data structures and algorithms that have fast query time while using linear or near-linear space for a variety of query shapes. In addition, a generic framework is presented, which returns a closest pair that is no farther apart than the closest pair in a suitably shrunken version of the query range

Algorithmic embeddings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 233-242).We present several computationally efficient algorithms, and complexity results on low distortion mappings between metric spaces. An embedding between two metric spaces is a mapping between the two metric spaces and the distortion of the embedding is the factor by which the distances change. We have pioneered theoretical work on relative (or approximation) version of this problem. In this setting, the question is the following: for the class of metrics C, and a host metric M', what is the smallest approximation factor a > 1 of an efficient algorithm minimizing the distortion of an embedding of a given input metric M E C into M'? This formulation enables the algorithm to adapt to a given input metric. In particular, if the host metric is "expressive enough" to accurately model the input distances, the minimum achievable distortion is low, and the algorithm will produce an embedding with low distortion as well. This problem has been a subject of extensive applied research during the last few decades. However, almost all known algorithms for this problem are heuristic. As such, they can get stuck in local minima, and do not provide any global guarantees on solution quality. We investigate several variants of the above problem, varying different host and target metrics, and definitions of distortion.(cont.) We present results for different types of distortion: multiplicative versus additive, worst-case versus average-case and several types of target metrics, such as the line, the plane, d-dimensional Euclidean space, ultrametrics, and trees. We also present algorithms for ordinal embeddings and embedding with extra information.by Mihai Bădoiu.Ph.D