Algorithmic Aspects of the Internet
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The goal of this thesis is to use and advance the techniques developed in the field of exact and approximation algorithms for many of the problems arising in the context of the Internet. We will formalize the method of dual fitting and the idea of factor-revealing LP. We use this combination to design and analyze two greedy algorithms for the metric uncapacitated facility location problem. Their approximation factors are 1.861 and 1.61 respectively. We also provide the first polynomial time algorithm for the linear version of a market equilibrium model defined by Irving Fisher in 1891. Our algorithm is modeled after Kuhn's primal-dual algorithm for bipartite matching. We also study the connectivity properties of the Internet graph and its impact on its structure. In particular, we consider the model of growth with preferential attachment for modeling the graph of the Internet and prove that under some reasonable assumptions, this graph has a constant conductance.