Subset feedback vertex set in tournaments as fast as without the subset

Abstract

In the FEEDBACK VERTEX SET IN TOURNAMENTS (FVST) problem, we are given a tournament T and a positive integer k. The objective is to determine whether there exists a vertex set X⊆V(T) of size at most k such that T−X is a directed acyclic graph. This problem is known to be equivalent to the problem of hitting all directed triangles, thereby using the best-known algorithm for the 3-HITTING SET problem results in an algorithm for FVST with a running time of 2.076k⋅nO(1) [Wahlström, Ph.D. Thesis]. Kumar and Lokshtanov [STACS 2016] designed a more efficient algorithm with a running time of 1.6181k⋅nO(1). A generalization of FVST, called SUBSET-FVST, includes an additional subset S⊆V(T) in the input. The goal for SUBSET-FVST is to find a vertex set X⊆V(T) of size at most k such that T−X contains no directed cycles that pass through any vertex in S. This generalized problem can also be represented as a 3-HITTING SET problem, leading to a running time of 2.076k⋅nO(1). Bai and Xiao [Theoretical Computer Science 2023] improved this and obtained an algorithm with running time 2k+o(k)⋅nO(1). In our work, we extend the algorithm of Kumar and Lokshtanov [STACS 2016] to solve SUBSET-FVST, obtaining an algorithm with a running time 1.6181k+nO(1), matching the running time for FVST. © 2026 Elsevier Inc.