Parameterized Complexity of the Sparsest k-Subgraph in Chordal Graphs

Overview & Synopsis

(This is the long version of the paper accepted to SOFSEM 2014) In this paper we study the Sparsest k-Subgraph problem which consists in finding a subset of k vertices in a graph which induces the minimum number of edges. The Sparsest k-Subgraph problem is a natural generalization of the Independent Set problem, and thus is NP-hard (and even W[1]-hard) in general graphs. In this paper we investigate the parameterized complexity of both Sparsest k-Subgraph and Densest k-Subgraph in chordal graphs. We first provide simple proofs that Densest k-Subgraph in chordal graphs is FPT and does not admit a polynomial kernel unless NP ⊆ coNP/poly (both parameterized by k). More involved proofs will ensure the same behavior for Sparsest k- Subgraph in the same graph class. We lastly provide an FPT algorithm in interval graphs for Sparsest k-Subgraph, but parameterized by the number of edges of the solution (a stronger parameterization than by k).