Quick-Sort Style Approximation Algorithms for Generalizations of Feedback Vertex Set in Tournaments

TL;DR

The paper generalizes the fast, randomized, factor-2 approach of quicksort-style approximations for FVST in tournaments to two broader settings: (i) DFVS-bIN in digraphs with independence number $α$, achieving a randomized $2α$-approximation in time $n^{\mathcal{O}(α^2)}$, and (ii) S-FVST in tournaments, achieving a randomized $2$-approximation in polynomial time. The core technique introduces an HL-degree ordering and pivot-based decomposition around a random vertex, coupled with local-ratio weight updates, to reduce to smaller subproblems while preserving approximation guarantees. For S-FVST, the authors adapt the framework to hit all cycles through a terminal set $S$, leveraging triangle-based reductions and vertex-cover tools for a tractable base case, yielding a 2-approximation with polynomial time. Together, these results extend the fast, probabilistic paradigm for cycle-hitting in dense digraphs and recover the tournament case ($α=1$) and the full-terminal subset variant as special cases, broadening the applicability of quicksort-style approximation techniques in combinatorial optimization on digraphs.

Abstract

A feedback vertex set (FVS) in a digraph is a subset of vertices whose removal makes the digraph acyclic. In other words, it hits all cycles in the digraph. Lokshtanov et al. [TALG '21] gave a factor 2 randomized approximation algorithm for finding a minimum weight FVS in tournaments. We generalize the result by presenting a factor $2α$ randomized approximation algorithm for finding a minimum weight FVS in digraphs of independence number $α$; a generalization of tournaments which are digraphs with independence number $1$. Using the same framework, we present a factor $2$ randomized approximation algorithm for finding a minimum weight Subset FVS in tournaments: given a vertex subset $S$ in addition to the graph, find a subset of vertices that hits all cycles containing at least one vertex in $S$. Note that FVS in tournaments is a special case of Subset FVS in tournaments in which $S = V(T)$.

Theorems & Definitions (20)

• theorem 1.1
• theorem 1.2
• lemma thmcounterlemma: seymour15
• lemma thmcounterlemma: $\dagger$
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma: $\dagger$
• definition thmcounterdefinition
• Claim 1.3: $\dagger$
• proof