A deterministic $(2 + \varepsilon)$-approximation for directed feedback vertex sets in tournaments
Abstract: We nearly settle the polynomial-time approximability of the Directed Feedback Vertex Set problem in tournaments. This problem is Vertex Cover-hard, and thus cannot have a $(2 - \varepsilon)$-approximation for any $\varepsilon > 0$ in polynomial time assuming the Unique Games Conjecture. In the past 28 years, several works have attempted to attain this approximability barrier of 2, and have designed algorithms with smaller and smaller approximation factors. This includes a $5/2$-approximation by Cai, Deng and Zang (FOCS 1998, SICOMP 2001); a $7/3$-approximation by Mnich, Vassilevska Williams and V{é}gh (ESA 2016), another $7/3$-approximation by Aprile, Drescher, Fiorini and Huynh (DAM 2023), and a $9/4$-approximation by Ghorbani and Mnich (ICALP 2026). Our main result improves upon all of those works: we give the first deterministic polynomial-time $(2+\varepsilon)$-approximation for Directed Feedback Vertex Set in tournaments, for all $\varepsilon > 0$. We thereby almost answer an open question by Lokshtanov, Misra, Mukherjee, Panolan, Philip and Saurabh (SODA 2020) who asked for a deterministic 2-approximation in polynomial time. Furthermore, we extend our result to the broader class of quasi-transitive digraphs