Approximating the Minimum Chain Completion problem

A bipartite graph G = (U, V, E) is a chain graph [9] if there is a bijection π : {1, . . . , |U |} → U such that Γ (π (1)) ⊇ Γ (π (2)) ⊇ . . . ⊇ Γ (π (|U |)), where Γ is a function that maps a node to its neighbors. We give approximation algorithms for two variants of the Minimum Chain Completion problem, where we are given a bipartite graph G(U, V, E), and the goal is find the minimum set of edges F that need to be added to G such that the bipartite graph G′ = (U, V, E′) (E′ = E ∪ F ) is a chain graph.