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)nO(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 2O(k 2)nO(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. ∗Supported by Pareto-Optimal Parameterized Algorithms, ERC Starting Grant 715744 and Parameterized Approximation, ERC Starting Grant 306992. M. S. Ramanujan also acknowledges support from BeHard, Bergen Research Foundation and X-Tract, Austrian Science Fund (FWF, project P26696). †University of Bergen, Bergen, Norway. [email protected] ‡Algorithms and Complexity Group, TU Wien, Vienna, Austria. [email protected] ‡The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected] §University of Bergen, Bergen, Norway. [email protected] ar X iv :1 70 4. 04 24 9v 1 [ cs .D S] 1 3 A pr 2 01 7