Parameterized complexity of computing maximum minimal blocking and hitting sets

02/05/2021
by   Júlio Araújo, et al.
0

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 parameterized by the distance to a graph class F. Indeed, it turns out that the existence of a polynomial kernel for this problem is closely related to the property that mmbs( F)=sup_G ∈ F mmbs(G) is bounded by a constant, and thus several recent results focused on determining mmbs( F) for different classes F. We consider the parameterized complexity of computing mmbs under various parameterizations, such as the size of a maximum independent set of the input graph and the natural parameter. We provide a panorama of the complexity of computing both mmbs and mmhs, which is the size of a maximum minimal hitting set of a hypergraph, a closely related parameter. Finally, we consider the problem of computing mmbs parameterized by treewidth, especially relevant in the context of kernelization. Given the "counting" nature of mmbs, it does not seem to be expressible in monadic second-order logic, hence its tractability does not follow from Courcelle's theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/09/2019

Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynom...
research
08/03/2022

Maximum Minimal Feedback Vertex Set: A Parameterized Perspective

In this paper we study a maximization version of the classical Feedback ...
research
02/04/2022

Globally Minimal Defensive Alliances: A Parameterized Perspective

A defensive alliance in an undirected graph G=(V,E) is a non-empty set o...
research
03/14/2021

Computing the Multicover Bifiltration

Given a finite set A⊂ℝ^d, let Cov_r,k denote the set of all points withi...
research
07/02/2023

Ranked Enumeration of Minimal Separators

Let G be an undirected graph, and s,t distinguished vertices of G. A min...
research
11/01/2022

A Near-Linear Kernel for Two-Parsimony Distance

The maximum parsimony distance d_MP(T_1,T_2) and the bounded-state maxim...
research
01/03/2020

On Supergraphs Satisfying CMSO Properties

Let CMSO denote the counting monadic second order logic of graphs. We gi...

Please sign up or login with your details

Forgot password? Click here to reset