Approximating element-weighted vertex deletion problems for the complete k-partite property

A k-partite graph is a graph G = (V₁,...,V_k, E), where V₁,...,V_k are k non-empty disjoint independent sets of vertices. Such a graph is called complete k-partite if E = U_{i ≠ j} V_i × V_j. 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.