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This paper concerns the problem of computing the densest k-vertex subgraph of a given graph, namely, the subgraph with the most edges, or with the highest edges-to-vertices ratio. A sequence of approximation algorithms is developed for the problem, with each step yielding a better ratio at the cost of a more complicated solution. The approximation ratio of our final algorithm is O/spl tilde/(n/sup 0.3885/). We also present a method for converting an approximation algorithm for an unweighted graph problem (from a specific class of maximization problems) into one for the corresponding weighted problem, and apply it to the densest subgraph problem.