Approximate Euclidean Steiner trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error e from 120 degrees. This notion arises in numerical approximations of minimum Steiner trees (W. D. Smith, Algorithmica, 7 (1992), 137–177). We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. Rubinstein, Weng and Wormald (J. Global Optim. 35 (2006), 573–592) conjectured that this relative error is at most linear in e, independent of the number of terminals. We verify their conjecture for the two-dimensional case as long as the error e is sufficiently small in terms of the number of terminals. We derive a lower bound linear in e for the relative error in the two-dimensional case when e is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error for larger values of e, and calculate exact values in the plane for three and four terminals

Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou

Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP

Given a metric space on n points, an {\alpha}-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an {\alpha} factor of the optimal Steiner tree cost for X. An {\alpha}-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an {\alpha} factor of the optimal as before, and satisfies the additional property that for any set X' that differs in a single vertex from X, the tree distributions for X and X' are "close" to each other. Universal and differentially private algorithms for TSP are defined similarly. An {\alpha}-approximate universal algorithm for the Steiner tree problem or TSP is also an {\alpha}-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(log n)-approximate differentially private algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root.Comment: 14 page