Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

TL;DR

This paper explores the fixed-parameter tractability of Vertex Cover and Feedback Vertex Set problems when parameterized by the difference between the solution size and various graph parameters, identifying the boundaries of tractability.

Contribution

It investigates the parameterizations of Vertex Cover and Feedback Vertex Set by the difference from multiple graph parameters, establishing new tractability boundaries.

Findings

Identifies parameters where the difference leads to FPT algorithms.

Shows certain parameterizations are W[1]-hard, indicating intractability.

Provides a comprehensive analysis of structural parameterizations.

Abstract

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem.