The DENSEST k-SUBGRAPH problem is a generalization of the maximum clique problem, in which we are given a graph G and a positive integer k, and we search among the subsets of k vertices of G one inducing a maximum number of edges. In this paper, we present algorithms for finding exact solutions of DENSEST k-SUBGRAPH improving the trivial exponential time complexity of O~* (2~n) and using polynomial space. Two FPT algorithms are also proposed; the first considers as parameter the treewidth of the input graph and uses exponential space, while the second is parameterized by the size of the minimum vertex cover and uses polynomial space. Finally, we propose several approximation algorithms running in moderately exponential or parameterized time.
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