Finding Long Directed Cycles Is Hard Even When DFVS Is Small Or Girth Is Large

TL;DR

This paper proves that finding long directed cycles remains computationally hard even when the directed feedback vertex set is small or the graph's girth is large, resolving open questions about the problem's complexity.

Contribution

It demonstrates W[1]-hardness of Hamiltonian Cycle and Longest Cycle when parameterized by directed feedback vertex set, even with large girth, and shows Longest Path is in XP.

Findings

Hamiltonian Cycle is W[1]-hard with small DFVS

Longest Cycle remains W[1]-hard when parameterized above girth

Longest Path is in XP when parameterized above girth

Abstract

We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization {\sc Longest Cycle}. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth} belongs to the class XP.