Approximation Algorithms for Maximization Problems Arising in Graph Partitioning

Given a graph G = (V, E), a weight function w: E → R⁺, and a parameter k, we consider the problem of finding a subset U  V of size k that maximizes: Max-Vertex Cover_k: the weight of edges incident with vertices in U,Max-Dense Subgraph_k: the weight of edges in the subgraph induced by U,Max-Cut_k: the weight of edges cut by the partition (U, VU),Max-Uncut_k: the weight of edges not cut by the partition (U, VU).For each of the above problems we present approximation algorithms based on semidefinite programming and obtain approximation ratios better than those previously published. In particular we show that if a graph has a vertex cover of size k, then one can select in polynomial time a set of k vertices that covers over 80% of the edges.