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2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (2011)
Palm Springs, California USA
Oct. 22, 2011 to Oct. 25, 2011
ISSN: 0272-5428
ISBN: 978-0-7695-4571-4
pp: 210-219
We obtain the first online algorithms for the node-weighted Steiner tree, Steiner forest and group Steiner tree problems that achieve a poly-logarithmic competitive ratio. Our algorithm for the Steiner tree problem runs in polynomial time, while those for the other two problems take quasi-polynomial time. Our algorithms can be viewed as online LP rounding algorithms in the framework of Buchbinder and Naor (Foundations and Trends in Theoretical Computer Science, 2009); however, while the {\em natural} LP formulation of these problems do lead to fractional algorithms with a poly-logarithmic competitive ratio, we are unable to round these LPs online without losing a polynomial factor. Therefore, we design new LP formulations for these problems drawing on a combination of paradigms such as spider decompositions, low-depth Steiner trees, generalized group Steiner problems, etc. and use the additional structure provided by these to round the more sophisticated LPs losing only a poly-logarithmic factor in the competitive ratio. As further applications of our techniques, we also design polynomial-time online algorithms with poly-logarithmic competitive ratios for two fundamental network design problems in edge-weighted graphs: the group Steiner forest problem (thereby resolving an open question raised by Chekuri et. al. (SODA 2008)) and the single source $\ell$-vertex connectivity problem (which complements similar results for the corresponding edge-connectivity problem due to Gupta et. al. (STOC 2009)).
Survivable Network Design, Online Algorithm

M. Singh, D. Panigrahi and J. (. Naor, "Online Node-Weighted Steiner Tree and Related Problems," 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science(FOCS), Palm Springs, California USA, 2011, pp. 210-219.
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