Approximation and Hardness Results for the Maximum Edges in Transitive Closure Problem

In this paper we study the following problem, named Maximum Edges in Transitive Closure, which has applications in computational biology. Given a simple, undirected graph G = (V, E) and a coloring of the vertices, remove a collection of edges from the graph such that each connected component is colorful (i.e., it does not contain two identically colored vertices) and the number of edges in the transitive closure of the graph is maximized. The problem is known to be APX-hard, and no approximation algorithms are known for it. We improve the hardness result by showing that the problem is NP-hard to approximate within a factor of vertical bar V vertical bar(1/3-epsilon), for any constant epsilon > 0. Additionally, we show that the problem is APX-hard already for the case when the number of vertex colors equals 3. We complement these results by showing the first approximation algorithm for the problem, with approximation factor root 2.OPT.