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 \(\ell \), and the goal is to find a vertex subset S of size at most \(\ell \) 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 \(n^{{\mathcal O}(k)}\). 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 \(\ell \), 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 \(f(k)\cdot n^{{\mathcal O}(1)}\), 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.
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Acknowledgment
We thank anonymous referees of an earlier version of the paper for several suggestions. Especially for finding a fatal flaw and giving suggestions for improving the running time of the algorithm.
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Abhinav, A., Bandopadhyay, S., Banik, A., Saurabh, S. (2022). Parameterized Algorithms for Finding Highly Connected Solution. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_1
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