true2016-03-16T16:38:26

We investigate the problem of converting sets of sensors into strongly connected networks of sensors using multiple directional antennae. Consider a set S of n points in the plane modeling sensors of an ad hoc network. Each sensor uses a fixed number, say 1 ≤ k ≤ 5, of directional antennae modeled as a circular sector with a given spread (or angle) and range (or radius). We give algorithms for orienting the antennae at each sensor so that the resulting directed graph induced by the directed antennae on the nodes is strongly connected. We also study trade-offs between the total angle spread and range for maintaining connectivity.

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We consider the conditional covering problem in an undirected network, in which each vertex represents a demand point that must be covered by a facility as well as a potential facility site. Each facility can cover all vertices within a given coverage radius, except the vertex at which the facility is located. The objective is to locate facilities to cover all vertices such that the total facility location cost is minimized. In this paper, new upper bounds are proposed for the conditional covering problem on paths, cycles, extended stars, and trees. In particular, we provide an O (n log n)-time algorithm for paths, an O (n 2 log n)-time algorithm for cycles, an O (n 1.5 log n)-time algorithm for extended stars, and an O (n 3 )-time algorithm for trees. Our algorithms for paths, extended stars, and trees improve the previous upper bounds from O (n 2 ), O (n 2 ), and O (n 4 ), respectively.

The paper provides a description of the two recent approximation algorithms for the Asymmetric Traveling Salesman Problem, giving the intuitive description of the works of Feige-Singh[1] and Asadpour et.al [2].[1] improves the previous O(log n) approximation algorithm, by improving the constant from 0.84 to 0.66 and modifying the work of Kaplan et. al [3] and also shows an efficient reduction from ATSPP to ATSP. Combining both the results, they finally establish an approximation ratio of 4 3 + ǫ log n for ATSPP, considering a small ǫ > 0, improving the work of Chekuri and Pal.[4] Asadpour et.al, in their seminal work [2], gives an O log n log log nrandomized algorithm for the ATSP, by symmetrizing and modifying the solution of the Held-Karp relaxation problem and then proving an exponential family distribution for probabilistically constructing a maximum entropy spanning tree from a spanning tree polytope and then finally defining the thin-ness property and transforming a thin spanning tree into an Eulerian walk. The optimization methods used in [2] are quite elegant and the approximation ratio could further be improved, by manipulating the thin-ness of the cuts.