Approximation Algorithms for Minimum Chain Vertex Deletion

A bipartite graph G = (A boolean OR B, E) is called a chain graph if there exists an ordering rho = < v(1), v(2), ... v(n) > of the vertices in A = {v(1), v(2), ... v(n)} such that N(v(1)) subset of N(v(2)) ... subset of N(v(2)). Here N(v) denotes the set of neighbors of the vertex v in G. We call the vertex-deletion problem corresponding to the class of chain graphs as the MINIMUM CHAIN VERTEX DELETION problem and the induced subgraph problem corresponding to chain graphs as the MAXIMUM INDUCED CHAIN SUBCRAPH problem. A weighted version of these problems is obtained by assigning positive weights on vertices and asking for a minimum weight deletion set to get into the class of chain graphs or asking for maximum weight induced chain subgraph. Using a rounding technique we show that the weighted version of MINIMUM CHAIN VERTEX DELETION, has a factor 2 approximation algorithm on bipartite graphs. We also give a factor 3/2 approximation algorithm for a weighted version of MAXIMUM INDUCED CHAIN SUBCRAPH on bipartite graphs. We also show that both these problems are APX-complete.