Parameterized Complexity and Approximability of Directed Odd Cycle Transversal

A directed odd cycle transversal of a directed graph (digraph) D is a vertex set S that intersects every odd directed cycle of D. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph D and an integer k. The objective is to determine whether there exists a directed odd cycle transversal of D of size at most k. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size k by showing that DOCT does not admit an algorithm with running time f (k)n O(1) unless FPT = W[1]. On the positive side, we give a factor 2 fixed parameter tractable (FPT) approximation algorithm for the problem. More precisely, our algorithm takes as input D and k, runs in time 2 O(k 2 ) n O(1) , and either concludes that D does not have a directed odd cycle transversal of size at most k, or produces a solution of size at most 2k. Finally, we provide evidence that there exists > 0 such that DOCT does not admit a factor (1 + ) FPT-approximation algorithm.