Approximating the Sparsest k-Subgraph in Chordal Graphs

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k -Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k -Subgraph is \(\mathcal{NP}\)-hard and even not approximable unless \(\mathcal{P} = \mathcal{NP}\) in general graphs. Thus, we investigate Sparsest k -Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the \(\mathcal{NP}\)-hardness of Sparsest k -Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a PTAS for Sparsest k -Subgraph on proper interval graphs.