Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Multivariate Approaches to Classification in Extragalactic Astronomy

Clustering objects into synthetic groups is a natural activity of any science. Astrophysics is not an exception and is now facing a deluge of data. For galaxies, the one-century old Hubble classification and the Hubble tuning fork are still largely in use, together with numerous mono-or bivariate classifications most often made by eye. However, a classification must be driven by the data, and sophisticated multivariate statistical tools are used more and more often. In this paper we review these different approaches in order to situate them in the general context of unsupervised and supervised learning. We insist on the astrophysical outcomes of these studies to show that multivariate analyses provide an obvious path toward a renewal of our classification of galaxies and are invaluable tools to investigate the physics and evolution of galaxies.Comment: Open Access paper. http://www.frontiersin.org/milky\_way\_and\_galaxies/10.3389/fspas.2015.00003/abstract\&gt;. \&lt;10.3389/fspas.2015.00003 \&g

Optimally Efficient Prefix Search and Multicast in Structured P2P Networks

Searching in P2P networks is fundamental to all overlay networks. P2P networks based on Distributed Hash Tables (DHT) are optimized for single key lookups, whereas unstructured networks offer more complex queries at the cost of increased traffic and uncertain success rates. Our Distributed Tree Construction (DTC) approach enables structured P2P networks to perform prefix search, range queries, and multicast in an optimal way. It achieves this by creating a spanning tree over the peers in the search area, using only information available locally on each peer. Because DTC creates a spanning tree, it can query all the peers in the search area with a minimal number of messages. Furthermore, we show that the tree depth has the same upper bound as a regular DHT lookup which in turn guarantees fast and responsive runtime behavior. By placing objects with a region quadtree, we can perform a prefix search or a range query in a freely selectable area of the DHT. Our DTC algorithm is DHT-agnostic and works with most existing DHTs. We evaluate the performance of DTC over several DHTs by comparing the performance to existing application-level multicast solutions, we show that DTC sends 30-250% fewer messages than common solutions

An oil pipeline design problem

Copyright @ 2003 INFORMSWe consider a given set of offshore platforms and onshore wells producing known (or estimated) amounts of oil to be connected to a port. Connections may take place directly between platforms, well sites, and the port, or may go through connection points at given locations. The configuration of the network and sizes of pipes used must be chosen to minimize construction costs. This problem is expressed as a mixed-integer program, and solved both heuristically by Tabu Search and Variable Neighborhood Search methods and exactly by a branch-and-bound method. Two new types of valid inequalities are introduced. Tests are made with data from the South Gabon oil field and randomly generated problems.The work of the first author was supported by NSERC grant #OGP205041. The work of the second author was supported by FCAR (Fonds pour la Formation des Chercheurs et l’Aide à la Recherche) grant #95-ER-1048, and NSERC grant #GP0105574