A polylogarithmic approximation algorithm for group Steiner tree problem

The group Steiner tree problem is a generalization of the Steiner tree problem where we are given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimum-weight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as hard. We give a randomized $O(\log^3 n \log k)$-approximation algorithm for the group Steiner tree problem on an $n$-node graph, where $k$ is the number of groups.The best previous performance guarantee was $(1+\frac{\ln k}{2})\sqrt{k}$ (Bateman, Helvig, Robins and Zelikovsky). Noting that the group Steiner problem also models the network design problems with location-theoretic constraints studied by Marathe, Ravi and Sundaram, our results also improve their bicriteria approximation results. Similarly, we improve previous results by Slav{\'\i}k on a tour version, called the errand scheduling problem. We use the result of Bartal on probabilistic approximation of finite metric spaces by tree metrics to reduce the problem to one in a tree metric. To find a solution on a tree, we use a generalization of randomized rounding. Our approximation guarantees improve to $O(\log^2 n \log k)$ in the case of graphs that exclude small minors by using a better alternative to Bartal's result on probabilistic approximations of metrics induced by such graphs (Konjevod, Ravi and Salman) -- this improvement is valid for the group Steiner problem on planar graphs as well as on a set of points in the 2D-Euclidean case

Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Approximation algorithm for the group Steiner tree problem

Given a weighted graph with some subsets of vertices called groups, the group Steiner tree problem is to find a minimum-weight subgraph which contains at least one vertex from each group. We give a randomized algorithm with a polylogarithmic approximation guarantee for the group Steiner tree problem. The previous best approximation guarantee was O (i2k1/i) in time O (nik2i) (Charikar, Chekuri, Goel, and Guha). Our algorithm also improves existing approximation results for network design problems with location-based constraints and for the symmetric generalized traveling salesman proble

Algorithms for Optimizing Production DNA Sequencing

for the U.S. Department of Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. The Los Alamos National Laboratory strongly supports academic freedom and a researcher's right to publish � as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness

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Algorithms for Optimizing Production DNA Sequencing Eva Czabarka \Lambda

C. Torney Abstract We discuss the problem of optimally "finishing " a partially sequenced, reconstructed DNA segment. At first sight, this appears to be computationally hard. We construct a series of increasingly realistic models for the problem and show that all of these can in fact be solved to optimality in polynomial time, with near-optimal solutions available in linear time. Implementation of our algorithms could result in a substantial efficiency gain for automated DNA sequencing