# Parameterized Complexity and Approximability of Directed Odd Cycle Transversal

## TL;DR

This paper investigates the parameterized complexity of the Directed Odd Cycle Transversal problem, establishing its computational hardness and providing a 2-approximation FPT algorithm, along with evidence against better approximations.

## Contribution

It proves DOCT is W[1]-hard when parameterized by solution size and introduces a 2-approximation FPT algorithm, advancing understanding of its approximability.

## Key findings

- DOCT is W[1]-hard when parameterized by solution size.
- A 2-approximation FPT algorithm for DOCT is developed.
- No $(1+\epsilon)$-approximation FPT algorithm exists for some $\epsilon > 0$.

## Abstract

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 $\epsilon > 0$ such that DOCT does not admit a factor $(1+\epsilon)$ FPT-approximation algorithm.