Parameterized algorithms for finding highly connected solution

Abstract

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)⋅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.