Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Abstract

Vertex Cover parameterized by the solution size k is the quintessential fixedparameter 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.

A faster algorithm for Vertex Cover parameterized by solution size

“…In addition, there is an algorithm with runtime O * (3 k−(2 λ −M M ) ), where k − (2 λ −M M ) is the most stringent of the three above-guarantee parameters [GP16]. Recently, [KKK22] investigated other Above-Guarantee algorithms, in terms of other parameters such as chromatic number, degeneracy, etc.…”