Approximating Element-Weighted Vertex Deletion Problems for the Complete k-Partite Property

A k-partite graph is a graph G=(V1,?,Vk,E), where V1,?,Vk are k non-empty disjoint independent sets of vertices. Such a graph is called complete k-partite if E=?i?jVi×Vj. We discuss three variants of the following optimization problem: given a graph and a non-negative weight function on the vertices and edges, find a minimum weight set of vertices and incident edges whose removal from the graph leaves a complete k-partite graph. All the problems we consider are at least as hard to approximate as the weighted vertex cover problem.We use the local-ratio technique to develop 2-approximation algorithms for the first two variants of the problem. In particular, we present the first (linear time) 2-approximation algorithm for the edge clique complement problem. For other previously studied special cases our 2-approximation algorithms are better in terms of time complexity than the existing 2-approximation algorithms. We use approximation preserving reductions in order to (4?4/k)-approximate the third variant of the problem.