Sparsity in Covering Solutions

In the classical covering problems, the goal is to find a subset of vertices/edges that “covers” a specific structure of the graph. In this work, we initiate the study of the covering problems where given a graph G, in addition to the covering, the solution needs to be sparse, i.e., the number of edges with both the endpoints in the solution are minimized. We consider two well-studied covering problems, namely Vertex Cover and Feedback Vertex Set. In Sparse Vertex Cover, given a graph G, and integers k, t, the goal is to find a minimal vertex cover S of size at most k such that the number of edges in G[S] is at most t. Analogously, we can define Sparse Feedback Vertex Set. Both the problems are NP-hard. We studied these problems in the realm of parameterized complexity. Our results are as follows: Sparse Vertex Cover admits an O(k²) vertex kernel and an algorithm that runs in O(1.3953^k·n^O(1)) time.Sparse Feedback Vertex Set admits an O(k⁴) vertex kernel and an algorithm that runs in O(5^k·n^O(1)) time. Sparse Vertex Cover admits an O(k²) vertex kernel and an algorithm that runs in O(1.3953^k·n^O(1)) time. Sparse Feedback Vertex Set admits an O(k⁴) vertex kernel and an algorithm that runs in O(5^k·n^O(1)) time.