- Authors:
- U. Feige,
- M. Seltser
Given an n-vertex graph G and a parameter k, we are to find a k-vertex subgraph with the maximum number of edges. This problem is NP-hard. We show that the problem remains NP-hard even when the maximum degree in G is three. When G contains a k-clique, we give an algorithm that for any e < 0 finds a k-vertex subgraph with at least (1 - e)(k || 2) edges, in time nO((1 + logn/>sub<k>/sub<)/e). We study the applicability of semidefinite programming for approximating the dense k-subgraph problem. Our main result in this respect is negative, showing that for k @ n1/3, semidefinite programs fail to distinguish between graphs that contain k-cliques and graphs in which the densest k-vertex subgraph has average degree below logn.
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