TY - GEN

T1 - Sparsity in Covering Solutions

AU - Jain, Pallavi

AU - Rathore, Manveer Singh

N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

PY - 2024/1/1

Y1 - 2024/1/1

N2 - 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(1.3953k·nO(1)) time.Sparse Feedback Vertex Set admits an O(k4) vertex kernel and an algorithm that runs in O(5k·nO(1)) time. Sparse Vertex Cover admits an O(k2) vertex kernel and an algorithm that runs in O(1.3953k·nO(1)) time. Sparse Feedback Vertex Set admits an O(k4) vertex kernel and an algorithm that runs in O(5k·nO(1)) time.

AB - 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(1.3953k·nO(1)) time.Sparse Feedback Vertex Set admits an O(k4) vertex kernel and an algorithm that runs in O(5k·nO(1)) time. Sparse Vertex Cover admits an O(k2) vertex kernel and an algorithm that runs in O(1.3953k·nO(1)) time. Sparse Feedback Vertex Set admits an O(k4) vertex kernel and an algorithm that runs in O(5k·nO(1)) time.

KW - Feedback Vertex Set

KW - Kernelization

KW - Parameterized Complexity

KW - Sparsity

KW - Vertex Cover

UR - http://www.scopus.com/inward/record.url?scp=85188688166&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-55601-2_9

DO - 10.1007/978-3-031-55601-2_9

M3 - Conference contribution

AN - SCOPUS:85188688166

SN - 9783031556005

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 131

EP - 146

BT - LATIN 2024

A2 - Soto, José A.

A2 - Wiese, Andreas

PB - Springer Science and Business Media Deutschland GmbH

T2 - 16th Latin American Symposium on Theoretical Informatics, LATIN 2042

Y2 - 18 March 2024 through 22 March 2024

ER -