Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation

We present approximation algorithms for the metric uncapacitated facility location problem and the metric <italic>k</italic>-median problem achieving guarantees of 3 and 6 respectively. The distinguishing feature of our algorithms is their low running time: <italic>O(m</italic> log<italic>m</italic>) and <italic>O(m</italic> log<italic>m(L</italic> + log (<italic>n</italic>))) respectively, where <italic>n</italic> and <italic>m</italic> are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities. The main algorithmic ideas are a new extension of the primal-dual schema and the use of Lagrangian relaxation to derive approximation algorithms.

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