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 examine a family of maximization problems arising naturally when considering a subset U V of size exactly k. Speci cally we consider the problem of nding a subset U V of size k that maximizes : Max-k-Vertex Cover : the weight of edges incident with vertices in U . Max-k-Dense Subgraph : the weight of edges in the subgraph induced by U . Max-k-Cut : the weight of edges cut by the partition (U; V nU). Max-k-Not Cut : the weight of edges not cut by the partition (U; V n U). We present a number of approximation algorithms based on linear and semide nite programming, and obtain approximation ratios higher than those previously published.