Fixed-parameter tractability results for feedback set problems in tournaments

Abstract

Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and improve fixedparameter tractability results for these problems. We show that Feedback Vertex Set in tournaments (FVST) is amenable to the novel iterative compression technique, and we provide a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization. Moreover, we apply the iterative compression technique to d-Hitting Set, which generalizes Feedback Vertex Set in tournaments, and obtain improved upper bounds for the time needed to solve 4-Hitting Set and 5-Hitting Set. Using our parameterized algorithm for Feedback Vertex Set in tournaments, we also give an exact (not parameterized) algorithm for it running in O (1.709 n ) time, where n is the number of input graph vertices, answering a question of Woeginger [G.J. Woeginger, Open problems around exact algorithms, Discrete Appl. Math. 156 (3) (2008) 397-405].

Improved bounds for minimal feedback vertex sets in tournaments

“…This yields a branching vector of (7,7,5,3,5,7,4,3) This yields a branching vector of (7,7,7,3,1) This yields a branching vector of (5, 7, 7, 7, 5, 7, 1) that solves to 1.5793. Henceforth, we can assume that 2 , 3 have no common predecessor except possibly 1 .…”

“…This yields a branching vector of (4,4,4,4,4,4,6) that solves to 1.5903. Now suppose that 1 , 2 have a common predecessor and 3 is the predecessor of 3 .…”

“…In other words, a tournament is a digraph with exactly one arc between any two of its vertices. Various approaches have been suggested to solve the MINIMUM FVS problem on tournaments, including approximation algorithms [3,11], fixed-parameter algorithms [4,9] as well asalgorithm [6,13]. The running time of this approach is within a polynomial factor of the number ( ) of minimal FVS in .…”

Parameterizing Edge Modification Problems Above Lower Bounds

“…FAST is NP-complete [1] but fixed-parameter tractable with respect to k [2,14,16,20,27,42]. The running time of the current best fixed-parameter algorithm is 2 c· √ k + n O(1) where c ≤ 5.24 [20].…”

Fixed-Parameter Algorithms for Cluster Vertex Deletion

“…The related problem of making an antisymmetric relation transitive by omitting a minimum number of elements is also known as Feedback Vertex Set in tournaments (see, e. g., [11,34] for fixed-parameter tractability results).…”