Parameterized algorithms for finding highly connected solution

To introduce our question and the parameterization, consider the classical VERTEX COVER problem. In this problem, the input is a graph G on n vertices and a positive integer l, and the goal is to find a vertex subset S of size at most l such that G - S is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be n - k edge-connected. Clearly, the problem is NP-complete, as substituting k = n - 1, we obtain the CONNECTED VERTEX COVER problem. A simple observation also shows that the problem admits an algorithm with running time nO(k). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most l, such that G[S] is n - k edge-connected and G - S satisfies a property. We show that this version of well-known problems such as VERTEX COVER, FEEDBACK VERTEX SET, ODD CYCLE TRANSVERSAL and MULTIWAY CUT admit an algorithm with running time f (k) middot nO(1), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for n - k edge connected STEINER SUBGRAPH, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for MULTICUT.(c) 2022 Elsevier B.V. All rights reserved.