Parameterized Complexity of Computing Maximum Minimal Blocking and Hitting Sets

• Published in: Algorithmica Vol. 85; no. 2; pp. 444 - 491
• Main Authors: Araújo, Júlio, Bougeret, Marin, Campos, Victor A., Sau, Ignasi
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2023; Springer Nature B.V; Springer Verlag
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Complexity; Computation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Mathematics of Computing; Parameterization; Parameters; Theory of Computation; Maximum minimal blocking set; Vertex cover; Upper domination; Maximum minimal hitting set; Design and analysis of algorithms; Parameterized complexity; Treewidth; Kernelization; Fixed parameter tractability
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-01036-5

A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G . Let mmbs ( G ) be the size of a maximum (inclusion-wise) minimal blocking set of G . This parameter has recently played an important role in the kernelization of Vertex Cover with structural parameterizations. We provide a panorama of the complexity of computing mmbs parameterized by the natural parameter and the independence number of the input graph. We also consider the closely related parameter mmhs , which is the size of a maximum minimal hitting set of a hypergraph. Finally, we consider the problem of computing mmbs parameterized by treewidth, especially relevant in the context of kernelization. Since a blocking set intersects every maximum-sized independent set of a given graph and properties involving counting the sizes of arbitrarily large sets are typically non-expressible in monadic second-order logic, its tractability does not seem to follow from Courcelle’s theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem.